Robust Nash Equilibria and Second-Order Cone Complementarity Problems

Transcription

1 Robust Nash Equilibria and Second-Order Cone Complementarit Problems Shunsuke Haashi, Nobuo Yamashita, Masao Fukushima Department of Applied Mathematics and Phsics, Graduate School of Informatics, Koto Universit, Koto , Japan Abstract In this paper we consider a bimatrix game in which the plaers can neither evaluate their cost functions exactl nor estimate their opponents strategies accuratel. To formulate such a game, we introduce the concept of robust Nash equilibrium that results from robust optimiation b each plaer, and prove its existence under some mild conditions. Moreover, we show that a robust Nash equilibrium in the bimatrix game can be characteried as a solution of a second-order cone complementarit problem (SOCCP. Some numerical results are presented to illustrate the behavior of robust Nash equilibria. Kewords: bimatrix game; robust Nash equilibrium; second-order cone complementarit problem 1 Introduction We consider a bimatrix game where two plaers attempt to minimie their own costs. Let R n and R m denote strategies of Plaers 1 and 2, respectivel. Moreover, let Plaer 1 s cost function be given b f 1 (, := T A with cost matrix A R n m, and Plaer 2 s cost function be given b f 2 (, := T B with cost matrix B R n m. We suppose that the two plaers choose their strategies and from the nonempt closed convex sets S 1 R n and S 2 R m, respectivel. Then, the plaers determine their strategies b solving the following minimiation problems with the opponents strategies fixed: minimie T A subject to S 1, minimie T (1 B subject to S 2. A point (, satisfing argmin S1 T A and argmin S2 T B is called a Nash equilibrium [2]. Since the minimiation problems (1 are convex, the problem of finding a Nash equilibrium can be formulated as a variational inequalit problem (VIP [12]. Moreover, if S 1 and S 2 are given b S 1 = { R n g i (, i = 1,..., N} and S 2 = { R m h j (, j = 1,..., M} with some convex functions g i : R n R and h j : R m R, respectivel, then the VIP is further reformulated as a mixed complementarit problem (MCP, which is also called a box-constrained variational problem. Recentl, MCP has been extensivel studied and man efficient algorithms have been developed for solving it [7, 8, 25]. The concept of Nash equilibrium is premised on the accurate estimation of opponent s strateg and the exact evaluation of plaer s own cost function. Thus Nash equilibrium ma hardl represent the This research was supported in part b Grant-in-Aid for Scientific Research (B from Japan Societ for the Promotion of Science. address: {haashi, nobuo, fuku}@amp.i.koto-u.ac.jp 1

2 actual situation when those operations are subject to errors. To deal with such situations, we introduce the concept of robust Nash equilibrium, which is parallel to that of robust optimiation [3, 4, 5, 11]. In the field of game theor, there has been much stud on games with incomplete information and the robustness of equilibria. Harsani [14, 15, 16] defines a game with incomplete information as a game where each plaer s paoff function is given in a stochastic manner with its probabilit distribution. This is one of the most popular formulations of games with incomplete information. Kajii and Morris [2] adopt Harsani s formulation to define a concept of robust equilibria to incomplete information. Especiall, the show that games with strict equilibria do not necessaril have robust equilibria, and the unique correlated equilibrium of a game is robust. Moreover, the introduce the notion of p-dominance, and show that a p-dominant equilibrium is robust under an appropriate assumption. Ui [26] considers the robustness of equilibria of potential games, which is a class of games involving Monderer and Shaple s potential function [22]. He shows that the action profile uniquel maximiing a potential function is robust. Recentl, Morris and Ui [24] have unified the above discussions for p-dominant equilibria and equilibria of potential games. The define a generalied potential function that contains Monderer and Shaple s potential function [22] and Morris s characteristic potential function [23], and show that an action maximiing the generalied potential function is a robust equilibrium to incomplete information. In the above-mentioned references, the robustness means that an equilibrium is stable with respect to estimation errors. On the other hand, the robust Nash equilibrium introduced in this paper is an equilibrium that results from robust optimiation [3, 4, 5, 11] b each plaer. More precisel, our formulation is premised on the conditions (I (III in the next section. Indeed, these conditions are applicable to several actual problems such as dnamic economic sstems based on duopolistic competition with random disturbances, traffic equilibrium problems with incomplete information on travel costs, etc. We note that the concept of robustness in this paper is different from those considered in [2, 24, 26]. In what follows, we first define a robust Nash equilibrium for a bimatrix game, and discuss its existence. Then, we show that, under certain assumptions, the robust Nash equilibrium problem can be formulated as a second-order cone complementarit problem (SOCCP. The SOCCP is a class of complementarit problems where the complementarit condition is associated with the Cartesian product of second-order cones. Several methods for solving SOCCPs have been proposed recentl [6, 9, 13, 17, 18] Throughout the paper, we use the following notations. For a set X, P(X denotes the set of all subsets of X. For a function f : R n R m R, f(, : R n R and f(, : R m R denote the functions with and, respectivel, being fixed. R n + denotes the nonnegative orthant in R n, that is, R n + := {x R n x }. For a vector x R n, x denotes the Euclidean norm defined b x := x T x. For a matrix M R n m, M F denotes the Frobenius norm defined b M F := ( n mj=1 i=1 (M ij 2 1/2. I n R n n denotes the identit matrix, and e n R n denotes the vector of ones. For a matrix M R n m, Mi r denotes the i-th row vector and Mi c denotes the i-th column vector. 2 Robust Nash equilibria and its existence In this section, we define the robust Nash equilibrium of a bimatrix game, and give sufficient conditions for its existence. 2

3 Throughout the paper, we assume that the following three statements hold for each plaer i (i = 1, 2: (I (II (III Plaer 1 cannot estimate Plaer 2 s strateg exactl, but can onl estimate that it belongs to a set Z( R m containing. Similarl, Plaer 2 cannot estimate Plaer 1 s strateg exactl, but can onl estimate that it belongs to a set Y ( R n containing. Plaer 1 cannot estimate his/her cost matrix exactl, but can onl estimate that it belongs to a nonempt set D A R n m. Plaer 2 cannot estimate his/her cost matrix exactl, but can onl estimate that it belongs to a nonempt set D B R n m. Each plaer tries to minimie his/her worst cost under (I and (II. Now, we define the robust Nash equilibrium under the above three assumptions. To realie (III, we define functions f i : R n R m R (i = 1, 2 b { f 1 (, := max f 2 (, := max } Â D A, ẑ Z(, } T Âẑ { ŷ T ˆB ˆB DB, ŷ Y ( The functions f 1 (, and f 2 (, represent Plaer 1 s and Plaer 2 s worst costs, respectivel, under uncertaint as assumed in (I and (II. Plaers 1 and 2 then solve the following minimiation problems, respectivel: minimie f 1 (, subject to S 1, (3 minimie f 2 (, subject to S 2. Now, we are in a position to define the robust Nash equilibrium. Definition 1 Let functions f 1 and f 2 be defined b (2. If r argmin S1 f1 (, r and r argmin S2 f2 ( r,, that is, ( r, r is a Nash equilibrium of game (3, then ( r, r is called a robust Nash equilibrium of game (1. Next, we give a condition for the existence of a robust Nash equilibrium of game (1. Note that Y ( and Z( given in (I can be regarded as set-valued mappings. In what follows, we suppose that Y (, Z(, D A and D B in (I and (II satisf the following assumption. Assumption A. (2 (a (b Set-valued mappings Y : R n P(R n and Z : R m P(R m are continuous, and Y ( and Z( are nonempt compact for an R n and R m. D A R n m and D B R n m are nonempt and compact sets. The functions f 1 and f 2 defined b (2 are well-defined under this assumption. B simple arguments on continuit, we can show that f 1 and f 2 are continuous everwhere. Furthermore, we have the following lemma on the convexit of f 1 (, and f 2 (,. We omit the proof since it is trivial. Lemma 1 Suppose that Assumption A holds. Let f1 and f 2 be defined b (2. Then, for an fixed R m and R n, the functions f 1 (, and f 2 (, are convex. The next lemma is a fundamental result for noncooperative n-person game [2, Theorem 9.1.1]. 3

4 Lemma 2 Consider a noncooperative two-person game where cost functions are given b θ 1 : R n R m R and θ 2 : R n R m R. Suppose that functions θ 1 and θ 2 are continuous at an (,, and that functions θ 1 (, and θ 2 (, are convex. Suppose that S 1 and S 2 are nonempt compact convex sets. Then, the game has a Nash equilibrium. B the above lemmas, we obtain the following theorem for the existence of a robust Nash equilibrium of game (1 Theorem 1 Suppose that Assumption A holds, and that S 1 and S 2 are nonempt compact convex sets. Then, game (1 has a robust Nash equilibrium. Proof. B Lemma 1, the functions f 1 (, and f 2 (, are convex. Moreover, as pointed out earlier, f 1 and f 2 are continuous everwhere. Therefore, from Lemma 2, game (3 has a Nash equilibrium. This means, b Definition 1, that game (1 has a robust Nash equilibrium. 3 SOCCP formulation of robust Nash equilibrium In this section, we focus on the bimatrix game where each plaer takes a mixed strateg, that is, S 1 = {, e T n = 1} and S 2 = {, e T m = 1}, and show that the robust Nash equilibrium problem reduces to an SOCCP. The SOCCP is to find a vector (ξ, η, ζ R l R l R ν satisfing the conditions K ξ η K, G(ξ, η, ζ =, (4 where G : R l R l R ν R l R ν is a given function, ξ η denotes ξ T η =, and K is a closed convex cone defined b K = K l 1 K l 2 K lm with l j -dimensional second-order cones K l j = {(ζ 1, ζ 2 R R lj 1 ζ 2 ζ 1 }. Since K 1 is the set of nonnegative reals, the nonlinear complementarit problem (NCP is a special case of SOCCP (4 with K = K 1 K 1 and G(ξ, η, ζ := F (ξ η for a given function F : R l R l. Consider the bimatrix game where Plaers 1 and 2 solve the following minimiation problems (5 and (6, respectivel: minimie minimie T A subject to, e T n = 1, (5 T B subject to, e T m = 1. (6 It is well known that a Nash equilibrium of this game is given as a solution of a mixed linear complementarit problem. In fact, since and are fixed in (5 and (6, respectivel, both problems are linear programming problems, and their KKT conditions are given b A + e n s, e T n = 1, B T + e m t, e T m = 1, (7 where s R and t R are Lagrange multipliers associated with the equalit constraints in (5 and (6, respectivel. Thus, if some (, satisfies the above two KKT conditions simultaneousl, then it is a Nash equilibrium of the bimatrix game. The problem of finding such a (, can be further formulated as a linear complementarit problem (LCP [1] 4

5 Now, we consider bimatrix games involving several tpes of uncertaint, and show that the robust Nash equilibrium problem corresponding to each game reduces to an SOCCP of the form K Mζ + q Nζ + r K, Cζ = d (8 with variable ζ R l+τ and constants M, N R l (l+τ, q, r R l, C R τ (l+τ and d R τ. Note that, b introducing new variables ξ R l and η R l, problem (8 reduces to SOCCP (4 with ν = l+τ and G : R 3l+τ R 2l+τ defined b G(ξ, η, ζ := 3.1 Uncertaint in the opponent s strateg ξ Mζ q η Nζ r. Cζ d In this subsection, we consider the case where each plaer estimates the cost matrix exactl but opponent s strateg uncertainl. More specificall, we make the following assumption. Assumption 1 (a Y ( := { + δ R n δ ρ, e T n δ = } and Z( := { + δ Rm δ ρ, e T mδ = }, where ρ and ρ are given positive constants. (b D A = {A} and D B = {B}, where A R n m and B R n m are given constant matrices. Here, the conditions e T n δ = e T mδ = in the definitions of Y ( and Z( are provided so that e T n ( + δ = et m ( + δ = 1 holds from et n = et m = 1. Under this assumption, the following theorem holds. Theorem 2 If Assumption 1 holds, then the bimatrix game has a robust Nash equilibrium. Proof. It is easil seen that Assumptions 1(a and 1(b impl Assumptions A(a and A(b, respectivel. Hence, the theorem readil follows from Theorem 1. We now show that the robust Nash equilibrium problem can be formulated as SOCCP (8 under Assumption 1. Plaer 1 solves the following minimiation problem to determine his/her strateg: { } minimie max T A( + δ δ ρ, e T m δ = subject to e T n = 1,. (9 Since the projection of vector A T onto hperplane π := { e T m = } can be represented as (I m m 1 e m e T m AT, the cost function can be written as { } f 1 (, = max T A( + δ δ ρ, e T mδ = { } = T A + max T Aδ δ ρ, e T m δ = = T A + ρ ÃT, where à := A(I m m 1 e m e T m. Hence, b introducing an auxiliar variable R, problem (9 can be reduced to the following convex minimiation problem: minimie, T A + ρ subject to ÃT,, e T n = 1. 5

6 This is a second-order cone programming problem [1, 21] and its KKT conditions can be written as the following SOCCP: ( ( ( K m+1 λ 1 λ Ã T K m+1, R n + A Ãλ + e ns R n +, et n = 1, λ = ρ, where λ R m and s R are Lagrange multipliers, and λ R is an auxiliar variable. In a similar manner, the KKT conditions for Plaer 2 can be written as ( ( ( K n+1 µ 1 K µ B n+1, R m + B T B T µ + e m t R m +, e T m = 1, µ = ρ, where µ R n and t R are Lagrange multipliers, and µ R is an auxiliar variable. Consequentl, the problem to find (, satisfing the above two KKT conditions simultaneousl can be reformulated as SOCCP (8 with l = 2m + 2n + 2, τ = 4, K = K n+1 K m+1 R m + Rn +, ζ = N = C = λ λ µ µ s t, M = 1 I m 1 I n I n I m 1 Ã T 1 B A Ã e n B T B T e m e T n e T m Component-wise uncertaint in the cost matrices, q =,, r =, 1, d = 1 ρ. ρ In the following three subsections 3.2, 3.3 and 3.4, we consider the case where each plaer estimates the opponent s strateg exactl but his/her own cost matrix uncertainl. In this subsection, we particularl focus on the case where the uncertaint in each cost matrix occurs component-wise independentl. That is, we make the following assumption. Assumption 2 (a Y ( = {} and Z( = {}. 6

7 (b D A := {A + δa R n m δa ij (Γ A ij (i = 1,..., n, j = 1,..., m} and D B := {B + δb R n m δb ij (Γ B ij (i = 1,..., n, j = 1,..., m} with given constant matrices A R n m and B R n m and positive constant matrices Γ A R n m and Γ B R n m. Under this assumption, we have the following theorem. We omit the proof, since it is similar to that of Theorem 2. Theorem 3 If Assumption 2 holds, then the bimatrix game has a robust Nash equilibrium. From Assumption 2, together with the constraints and, the cost function f 1 can be represented as { } f 1 (, = max T Â Â D A n m = T A + max δa ij i j δa ij (Γ A ij i=1 j=1 n m = T A + (Γ A ij i j i=1 j=1 = T (A + Γ A. Analogousl, we have f 2 (, = T (B + Γ B. Hence, the robust Nash equilibrium problem is simpl the problem of finding a Nash equilibrium of the bimatrix game with cost matrices A+Γ A and B +Γ B. This problem reduces to the MCP (7 with A and B replaced b A + Γ A and B + Γ B, respectivel. 3.3 Column/row-wise uncertaint in the cost matrices In this subsection, we focus on the case where the uncertaint in matrices A and B respectivel occur row-wise independentl and column-wise independentl. That is, we make the following assumption. Assumption 3 (a Y ( = {} and Z( = {}. (b D A := {A + δa δa c j (γ A j (j = 1,..., m} and D B := {B + δb δb r i (γ B i (i = 1,..., n} with given constant matrices A R n m and B R n m and positive constant vectors γ A R m and γ B R n. This assumption implies that the degree of uncertaint in each plaer s cost depends on the opponent s each pure strateg. Under this assumption, we have the following theorem. We omit the proof, since it is similar to that of Theorem 2. Theorem 4 If Assumption 3 holds, then the bimatrix game has a robust Nash equilibrium. Next, we formulate the Nash equilibrium problem as an SOCCP. From Assumption 3, we have f 1 (, = max T Â Â D A = T A + max m j T δa c j δa c j (γ A j j=1 m = T A + j (γ A j j=1 = T A + γ T A 7

8 and γ T A. Hence, b introducing an auxiliar variable R, Plaer 1 s problem can be written as minimie, T A + (γ T A subject to,, e T n = 1. This is a second-order cone programming problem, and its KKT conditions are given b ( ( K n+1 γa T K n+1 A + e n s λ R n + λ Rn +, et n = 1 with Lagrange multipliers λ R n and s R. In a similar wa, the KKT conditions for Plaer 2 s minimiation problem are given b ( ( K m+1 γb T B T K m+1 + e m t µ R m + µ Rm +, et m = 1, where µ R m and t R are Lagrange multipliers. Combining the above two KKT conditions, we obtain SOCCP (8 with l = 2m + 2n + 2, τ = 2, K := K n+1 K m+1 R n + Rm +, 1 I n ζ = 1 λ, M =, q =, µ I m I n s I m t γ T A A I n e n γb T N = B T, r =, I m e m I n I m C = ( e T n e T m 3.4 General uncertaint in the cost matrices, d = ( 1. 1 In this subsection, we consider the general case of uncertaint in each cost matrix. That is, we make the following assumption. Assumption 4 (a (b Y ( = {} and Z( = {}. D A := {A + δa R n m δa F ρ A } and D B := {B + δb R n m δb F ρ B } with given constant matrices A R n m and B R n m and positive scalars ρ A and ρ B. 8

9 Under this assumption, we also have the following theorem. Theorem 5 If Assumption 4 holds, then the bimatrix game has a robust Nash equilibrium. Next, we consider the SOCCP formulation of the game. First note that Moreover, we have f(, = max{ T Â Â D A} = T A + max T (δa. δa F ρ A max T (δa = δa F ρ A max ( T vec(δa = ρ A = ρ A, δa F ρ A where vec( denotes the vec operator that creates an nm-dimensional vector ((p c 1 T,..., (p c m T T from a matrix P R n m with column vectors p c 1,..., pc m, and denotes Kronecker product (see Sections 4.2 and 4.3 in [19] Hence, b introducing an auxiliar variable R, Plaer 1 s minimiation problem reduces to the following problem: minimie, T A + ρ A subject to, e T n = 1,. Again, this is a second-order cone programming problem, and its KKT conditions are given b ( ( K n+1 ρ A K n+1, A + e n s λ R n + λ R n + e T n = 1, where λ R n and s R are Lagrange multipliers. Moreover, we can show that this SOCCP is rewritten as the following SOCCP: ( ( K n+1 ρ A 1 K n+1, e T n A + e n s λ = 1, ( ( K m+1 1 K m+1, R n + λ R n + (1 u with an auxiliar variable u R m. To see this, it suffices to notice that the complementarit condition ( ( K m+1 1 K m+1 (11 u in (1 implies = 1. This fact can be verified as follows: On one hand, ( 1 K m+1 implies 1. On the other hand, it holds that = 1 + T u 1 u 1, where the equalit follows from the perpendicularit in (11, the first inequalit follows from the Cauch-Schwar inequalit, and the last inequalit follows from the condition ( u K m+1 in (11. Moreover, e T n = 1 and ( K n+1 impl >. Hence, we have 1. In a similar wa, the KKT conditions for Plaer 2 s problem are given b ( ( K m+1 ρ B 1 K m+1, e T m = 1, K n+1 ( 1 B T + e m t µ ( v K n+1, R m + µ Rm +, 9

10 where t R, µ R m and v R n are auxiliar variables. Combining the above two KKT conditions, we obtain SOCCP (8 with l = 3m + 3n + 4, τ = 2, K = K n+1 K n+1 K m+1 K m+1 R n + R m +, ζ = N = C = 1 v 1 u λ µ s t, M = 1 I n 1 I n 1 I m 1 I m I n I m ρ A A I n e n 1 I n ρ B B T I n e m 1 I m I n I m ( ( e T n 1 e T, d =. m 1, q =,, r = As shown thus far, the robust Nash equilibrium problems under Assumptions 1, 3 and 4 are transformed into SOCCPs, while the robust Nash equilibrium problem under Assumption 2 reduces to an LCP. This is a natural consequence of the fact that the Euclidean norm is used in Assumptions 1, 3 and 4 to describe the uncertaint and the absolute value is used in Assumption 2. 4 Numerical examples of robust Nash equilibria In the previous section, we have shown that some robust Nash equilibrium problems for bimatrix games reduce to SOCCPs. In this section, we present some numerical examples for the robust Nash equilibria. Several methods have been proposed for solving SOCCPs. Among them, one of the most popular approaches is to reformulate the SOCCP as an equivalent nondifferentiable minimiation problem and solve it b Newton-tpe method combined with a smoothing technique [6, 9, 13, 18] In our numerical experiments, we use an algorithm based on the methods proposed in [18]. 4.1 Uncertaint in opponent s strateg We first stud the case where onl the opponents strategies involve uncertaint, that is, Assumption 1 holds. 1

11 We consider the bimatrix game with cost matrices: A 1 = , B 1 = (12 The Nash equilibrium of the game is given b = (.4815,.1852,.3333 and = (.1699,.2628, Robust Nash equilibria ( r, r for various values of (ρ, ρ are shown in Table 1, where f i ( r, r denotes the cost value of each plaer i = 1, 2 at a robust Nash equilibrium. From the table, we see that robust Nash equilibria ( r, r approach the Nash equilibrium (, as both ρ and ρ tend to. Note that Plaers 1 and 2 estimate the opponents strategies more precisel as ρ and ρ become smaller. However, the costs of both plaers for (ρ, ρ = (.5,.5 are smaller than those for (ρ, ρ = (.1,.1. This implies that an equilibrium ma not necessaril be favorable for either of the plaers even if the estimation is more precise. Next, we consider the bimatrix game with cost matrices: A 2 = , B 2 = (13 In this case, the Nash equilibrium comprises = (,, 1 and = (,, 1, and for an pair (ρ, ρ {.5,.1,.1} {.5,.1,.1}, robust Nash equilibrium ( r, r remains the same, that is, r = and r =. From this result, we ma expect that, if the Nash equilibrium is a pure strateg, then the robust Nash equilibrium remains unchanged even if there is uncertaint to some extent. 4.2 Uncertaint in cost matrices We next stud the general case where the plaers cost matrices involve uncertaint, that is, Assumption 4 holds. First we consider the bimatrix game with the cost matrices A 1 and B 1 defined b (12. Robust Nash equilibria for various values of (ρ A, ρ B are shown in Table 2, where f i ( r, r denotes the cost value of robust Nash equilibrium. As in the previous case, we see from the table that precise estimation for cost matrices does not necessaril reduce the cost at an equilibrium. Next we consider the bimatrix game with cost matrices A 2 and B 2 defined b (13. Robust Nash equilibria for various values of (ρ A, ρ B are shown in Table 3, which reveals that r = and r = hold when ρ A and ρ B are sufficientl small. We also see from the table that precise estimation for cost matrices does not alwas result in the reduction of the plaers costs at an equilibrium. 5 Concluding remarks In this paper, we have defined the concept of robust Nash equilibrium, and studied a sufficient condition for its existence. Moreover, we have shown that some robust Nash equilibrium problems can be reformulated as SOCCPs. To investigate the behavior of robust Nash equilibria, we have carried out some numerical examples. Our stud is still in the infanc, and man issues remain to be addressed. (1 One is to extend the concept of robust Nash equilibrium to the general N-person game. For the 2-person bimatrix game studied in this paper, it is sufficient to consider the uncertaint in the cost matrices and the 11

12 opponent s strateg. To discuss general N-person games, more complicated structure should be dealt with. (2 Another issue is to find other sufficient conditions for the existence of robust Nash equilibria. For instance, it ma be possible to consider the existence of robust Nash equilibria without assuming the boundedness of strateg sets. (3 Theoretical stud on the relation between Nash equilibrium and robust Nash equilibrium is also worthwhile. For example, it is not known whether the uniqueness of Nash equilibrium is inherited to robust Nash equilibrium. (4 In this paper, we have formulated several robust Nash equilibrium problems as SOCCPs. However, we have onl considered the cases where either the cost matrices or the opponent s strateg is uncertain for each plaer. It seems interesting to stud the case where both of them are uncertain, or the structure of uncertaint is more complicated. (5 In our numerical experiments, we emploed an existing algorithm for solving SOCCPs. But, there is room for improvement of solution methods. It ma be useful to develop a specialied method for solving robust Nash equilibrium problems. Table 1: Robust Nash equilibria for various values of ρ and ρ ρ ρ r r f 1 ( r, r f 2 ( r, r.1.1 (.4896,.1814,.329 (.172,.2697, (.563,.1482,.2888 (.1758,.334, (.5621,.156,.2819 (.1948,.632, (.8891,.11,.198 (.1812,.3272, (.884,.432,.729 (.2129,.5929, Table 2: Robust Nash equilibria for various values of ρ A and ρ B (cost matrices A 1 and B 1 ρ A ρ B r r f 1 ( r, r f 2 ( r, r.1.1 (.4841,.1797,.3362 (.1721,.2623, (.597,.1376,.3527 (.1969,.2552, (1.,.,. (.2931,.2326, (.583,.195,.2967 (.3497,.2453, (.5934,.1961,.215 (.3326,.32, Table 3: Robust Nash equilibria for various values of ρ A and ρ B (cost matrices A 2 and B 2 ρ A ρ B r r f 1 ( r, r f 2 ( r, r.1.1 (.,., 1. (.,., (.,., 1. (.,., (.,., 1. (.,.311, (.,.4286,.5714 (.,., (.,.3783,.6217 (.,.1935,

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