Generalized Colonel Blotto Game

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(i.e. the [9] a strategies of in in deterministic work equilibrium in seminal the equilibrium the of instance, existence For the analyzing on utpepaes[] n eeoeeu eore [11]–[1 resources heterogeneous and [4], resource players those symmetric multiple including [10], [8], studied valuations been homogeneous have with CBG the of variants many h ancnrbto fti ae steeoet develop to therefore is paper this of contribution main The case the been has as strategies, mixed on relying However, focused primarily has literature of body existing This eeaie ooe ltoGame Blotto Colonel Generalized 1 n ae Bas¸ar Tamer and , 2 ( GCBG which ) egies) umes bility urce ver 3]. nd ve m c, s, t, n f - r t hence, win the game. In contrast with the classical CBG, in For the CBG, it has been proven [9] that for Rb > Ra and our proposed GCBG: nRa > Rb, there exists no pure-strategy NE. As a result, • We derive deterministic equilibrium strategies which for solving the CBG, a common methodology is to explore are practical, in the sense that the players can derive mixed strategies based on which each player j chooses a ∈P exact optimal strategies and do not need to randomize probability distribution (or a multi-variate probability density between possible strategies, function) over j to maximize its potential expected payoff. Q • We characterize low complexity solutions even for a However, the reliance on such mixed strategies introduces large number of battlefields with heterogeneous valua- high analytical complexity and presents challenges in terms tions, of the tractability of the derivation of the solutions as well • We model realistic applications of competitive resource as to the practicality and applicability of these solutions. allocation problems by taking into consideration partial Hence, to enable the derivation of tractable, deterministic, winning and losing over a battlefield. and practical solutions to the competitive resource allocation Finally, we provide a numerical example which showcases problem, we next propose a generalization of the CBG the effect of the number of resources of each player on the dubbed generalized Colonel Blotto game. NE strategies and outcomes of the GCBG. B. Proposed Generalized Colonel Blotto Game II. PROPOSED GENERALIZED COLONEL BLOTTO GAME j j n ˜ j A GCBG , j∈P , R j∈P , , vi i=1, U j∈P , A. Classical Colonel Blotto Game P {Q } { } N { } { } n j j n k is defined by seven components that include the players, A standard CBG [9] , ∈P , R ∈P , , v , P {Q }j { }j N { i}i=1 strategy spaces, available resources, battlefields, and valua- j o U j∈P is defined byn six components: a) the players in the tion of battlefields as is the case in the CBG. However, we { } ˜ j set , , b) the strategy spaces j for , c) the define a new utility function U j∈P as an approximation to oa,b j { } availableP { resource} Rj for j , d) aQ set of n ∈Pbattlefields, the utility function of the CBG. This approximation is based ∈ P , e) the normalized value of each battlefield, vi for i , on a continuous differentiable function and an approximation andN f) the utility function, U j, for each player. The set∈ N of parameter, k. When k increases, the GCBG will very closely possible strategies for each player, j , corresponds to the resemble the CBG. At the limit, when k goes to infinity, the different possible distributions of its QRj resources across the GCBG converges to the CBG. However, in contrast with the battlefields. Hence, j is defined as CBG, the differentiability of the approximate utility function Q n enables the derivation of deterministic equilibrium strategies. j = rj rj Rj , rj 0 , (1) In this respect, as defined in (2), the payoff from each Q i ≤ i ≥ ( i=1 ) battlefield resembles a step function. Hence, we propose an X − j j j j where r is the number of allocated resource units by player approximation of this function, u˜i (ri , ri , k), referred to i j to battlefield i, and rj j is the vector of these allocated hereinafter as k approximation, defined as follows: ∈Q − resources (an allocation strategy of player j) . To this end, − vi − vi u˜j (rj , r j , k) , arctan k(rj r j ) + , (5) the payoff of player j, for a battlefield i, is defined as: i i i π i − i 2 − j j j vi, if ri > ri , and the utility function of each player U˜ is the summation − − uj(rj , r j )= vi , if rj = r j , (2) of the k approximation payoffs from each battlefield. Next, i i i  2 i i −  j −j in Lemma 1, we will show that (5) precisely approximates 0, if ri < ri , uj (rj , r−j ) as the approximation factor k tends to infinity. −j i i i where ri corresponds to the resources allocated by the Fig. 1 provides a number of plots of k approximation opponent of j to battlefield i. Such resource allocation functions for different values of k. − strategies are known as pure deterministic strategies since Lemma 1. The payoff from each battlefield i can be the allocation is not randomized over the battlefields. The ∈ N utility function of each player, U j , over all battlefields in expressed as follows: N is defined as: j j −j j j −j n ui (ri , ri ) = lim u˜i (ri , ri , k). (6) j j −j j j −j k→∞ U (r , r )= ui (ri , ri ). (3) i=1 Proof. The limit of the k-approximate payoff when k X → ∞ Each player in aims to choose a resource allocation can be written as: P − strategy that maximizes its payoff over the n battlefields vi arctan(+ )+ vi = vj , rj > r j , π ∞ 2 i i i given the potential resource allocation strategy of its op- j j −j vi vi j −j lim u˜i (ri , ri , k)=0+ 2 = 2 , ri = ri , ponent. As such, the pure-strategy NE of the CBG can be k→∞ −  vi arctan( )+ vi =0, rj < r j . formally defined as follows:  π 2 i i ∗ ∗ ∗ −∞ Definition 1. A strategy profile (rj , r−j ), such that rj Therefore, as compared to (2), j −j ∗ −j ∈  and r , is a pure-strategy Nash equilibrium of j j −j j j −j Q ∈Q lim u˜i (ri , ri , k)= ui (ri , ri ), (7) the CBG if k→∞ ∗ ∗ ∗ U j (rj , r−j ) U j (rj , r−j ), rj j . (4) which proves the lemma. ≥ ∀ ∈Q 0.5 rb b. Given the resource limitation of each player, a ∈ Q 0.4 limit exists on the chosen allocations as expressed in (1). First, we prove that this limit is binding, i.e. each player is 0.3 always better off spending all their available resources. n j j 0.2 Proposition 1. All the strategies of the form i=1 ri < R n j j are dominated by the strategies i=1 ri = R i.e, player j 0.1 is better off using all of its available resource.P P 0 Proof. The proof is by contradiction. Suppose r¯j = -10 -8 -6 -4 -2 0 2 4 6 8 10 j j ˜ j j −j [¯r1,..., r¯n] is the vector that maximizes U (r , r , k) such n j ¯j j that: i=1 r¯i = R < R . Then, the payoff for player j Fig. 1: k-approximate of the payoff from battlefield 1 for j a choosing strategy r¯ is: player a when v1 =0.5. P n ˜ j j −j vi j −j 1 Hence, Lemma 1 provides an approximation for the U (r¯ , r¯ , k)= arctan k(¯ri r¯i ) + . − π − 2 j j j i=1 payoff from each battlefield. Using u˜i (ri , ri , k) instead j j −j X of ui (ri , ri ) provides two additional benefits as com- Now, considering arbitrarily player a, we define a new a a a pared to the CBG: a) the continuity and differentiability of allocation vector for player a as ρ = [¯r1 + R j j −j ¯a a a − u˜i (ri , ri , k) helps in deriving pure-strategy NEs, and b) R , r¯2 ,..., r¯n]. Then, the payoff for player a will be: j j −j u˜ (r , r , k) captures the notion of partial-win-or-lose by v i i i U˜ a(ρa, r¯b, k)= 1 arctan k(¯ra + Ra R¯a r¯b) accounting for resource consumption and losses which is a π 1 − − i practical feature in a GCBG that has not been considered in n vi 1 the CBG. In fact, in practice, the payoff from each battlefield + arctan k(¯ra r¯b) + π i − i 2 must consider the portion of resources destroyed/consumed i=2 X following the “battle” between the two players. This means v n v 1 > 1 arctan k(¯ra r¯b) + i arctan k(¯ra r¯b) + that the payoff from each battlefield has to capture the loss π 1 − 1 π i − i 2 i=2 due to the resources destroyed in this battlefield, which also X ˜ a a b corresponds to a gain for the opponent. The k approximate = U (r¯ , r¯ , k), − utility function in (5) inherently captures this aspect. Fig. j j j which contradicts with the assumption r¯ = [¯r1,..., r¯n] 1 shows the continuous transition from losing to winning maximizes U˜ j (rj , r−j , k). a certain battlefield when using k-approximate payoff functions. Here, we note that the larger k is, the closer the k- From Proposition 1, we know that the limit on the total approximation approximates the classical CBG. number of allocated resources is binding. Before character- Now, we define the k-approximate utility function per izing the solution of the game, we first define the difference player: n between the allocated resources by each player on each − − ˜ j rj r j , j j j battlefield i , zi, and the difference between the total U ( , , k) u˜i (ri , ri , k). (8) ∈ N i=1 available resources, D, as follows: X z = ra rb,D = Ra Rb. (11) From (5) and (8), and given that arctan is an odd function, i i − i − the players’ payoffs from each battlefield can be related In addition, we consider, without loss of generality, that − − − j j j j j j b a following u˜i (ri , ri , k)+u ˜i (ri , ri , k) = vi, which R < R , which implies that the available resources of an results in the following relation between the players’ utility arbitrary player b are less than those of its opponent, player functions: a. We also consider the case in which Ra/n < Rb, which U˜ j (rj , r−j , k)+ U˜ −j (r−j, rj , k)=1. (9) eliminates the trivial case studied in the CBG, in which, if Ra > nRb, player a can trivially win the game. Using these The relation in (9) shows that the GCBG is a constant- two inequality assumptions on Ra and Rb, the constraint on sum game in which the pure-strategy NE is the solution of D can be expressed as follows: the following minimax problem [16]: 0 0. where

Proof. Based on Proposition 1, zn can be expressed as zn = vn ∗ n−1 z +z z z D z , and player a’s utility function will be: v1 1 n n n i=1 i ··· . − vn ∗ .  zn z2 +zn .  P n−1 H = v2 ··· . vi 1 . . . U˜ a(z, k)= arctan(kz ) . . .. . π i  . . . .  i=1  vn ∗  X  zn zn −1 zn−1+zn  n−1 n  ··· vn  vn vi  (25) + arctan k(D zi) + . (16) π − 2 ∗ i=1 ! i=1 For z to be a local maximum, H must be negative X X Hence, to find the local maxima of player a’s utility definite and, hence, H1 must be positive definite. Performing function, from the first order necessary condition of op- row operations on H1, we obtain an upper-triangular matrix: ∗ ∗ ∗ timality, we know that, if z is a local maximizer and z1 0 0 zn−1 ˜ a ˜ a ∗ 0 ··· . − . U is continuously differentiable, then U (z , k) = . ∗ .. . . ∇ ˜ a z 0 z . . . Therefore, for i , n , we must have ∂U ( ,k) =  2  ∈M N\{ } ∂zi . . . . 0, and, hence: H2 =  . .. .. 0 .  , (26)  .   . .. ∗ ∗  1 kvi 1 kvn  . . z z  =0,  n−2 n−1  2 2 2 n−1 2 − π k zi +1 − π k (D z ) +1  0 0 ξ  − l=1 l  ······  kv kv   i n (17) where 2 2 P 2 2 =0. k zi +1 − k zn +1 n−1 vn ∗ vizn ∗ ξ = zn−1 1+ ∗ . (27) Thus, zi must be one of two solutions: v − v z n 1 i=1 n i ! X 1 H is positive definite if and only if its pivots (i.e. its z∗ = (k2z2 v + v v ), i , (18) 1 i ± k2v n i i − n ∈M diagonal elements) are all positive since the number of r n n−1 ∗ positive pivots of H2 is equal to the number of positive which, from (13), must meet z + zn = D. i=1 i eigenvalues of H1 [17]. Hence, by inspecting the diagonal Next, from the second order necessary conditions of elements of H in (26), for H to be positive definite, P∗ 2 1 optimality, we know that, if z is a local maximizer of ∗ zi must be positive for i n 1 . Combining U˜ a(z, k), then the Hessian matrix, 2U˜ a(z∗, k), must be ∈ M\{ − } ∇ this result with (18), we obtain the expressions in (15) for negative definite. The second order derivatives, and elements i n 1 . of the Hessian matrix, are calculated as follows: ∈M\{Next, we− investigate} the sign of ξ in (27) which must 2 a 3 3 n−1 also be positive for H1 to be positive definite. Knowing that ∂ U˜ (z, k) 1 2k vizi 1 2k vn(D zl) = − l=1 z > 0 for i n 1 , we investigate four different 2 2 2 2 2 n−1 2 2 i ∂zi −π (k zi +1) −π (k (D zl) +1) ∈M\{ − }∗ l=1P cases regarding the signs of z − and zn and their effect on 3 −3 n 1 1 2k vizi 1 2k vnzn ∗ P (19) the sign of ξ. We prove that zn > 0 and zn−1 > 0 as defined = 2 2 2 2 2 2 , −π (k zi + 1) − π (k zn + 1) in (15) is the only case that leads to a positive definite H1 2 a 3 n−1 ∂ U˜ (z, k) 1 2k vn(D zl) and, as a result, a negative definite H. l=1 ∗ = −− Case 1. , and : If and , ∂z ∂z −π 2 n 1 2 2 zn−1 < 0 zn < 0 zn−1 < 0 zn < 0 i m (k (D l=1 zl) + 1) − P then the (n 1,n 1) element of H1 in (25) is negative. 1 2k3v z − − n nP (20) In addition, for a matrix to be positive definite, none of its = 2 2 2 . −π (k zn + 1) diagonal elements can be negative [17]. As such, if zn−1 < 0, ∗ and zn < 0, then H1 is not positive definite and hence H Therefore, the solutions z proposed in (15) and meet- is not negative definite. ing (14) such that zn > 0, are the only possible local maxima ∗ ˜ a Case 2. zn−1 < 0, and zn > 0: We rewrite ξ in (27) as of U (z, k). vn ∗ ∗ ξ = z − φ, where for z − < 0 and zn > 0, vn−1 n 1 n 1 In Theorem 1, we identified the local maxima of U˜ a(z, k) n−2 as given in (15) and meeting z > 0. We also proved that v z n φ =1+ i n these solutions must satisfy (14). Next, we study the equality 2 vi vi−vn i=1 vn z + 2 constraint in (14) which must be met. To this end, we let X n vn k vn n−1 qvn−1zn 1 f (z ) , z + (k2z2v + v v )= D. (33) − 2 vn−1 vn−1−vn k n n 2 n i i n vn z + 2 k vn − n vn k vn i=1 r X n−2q v z fk(zn) is a strictly convex function with a negative mini- =1+ i n vi vi−vn mizing zn as stated in Proposition 2. i=1 vnzn v 1+ k2v z2 X n i n Proposition 2. fk(zn) is a strictly convex function whose vn−1zn q q (28) minimum occurs at a negative zn. − vn−1 vn−1−vn · vnzn 1+ 2 2 vn k vn−1zn Proof. To prove the convexity of fk(zn), we compute its q q second derivative with respect to z . Based on the binomial approximation, (1 + ǫ)α 1+ αǫ n ≈ 2 2 2 n−1 1 2 2 vi 2 if ǫ < 1 and αǫ << 1, let ǫi = (vi vn)/(k vizn). d f 2 k znvi + vi vn zn | | | | − = k vn − − vn . (34) For practical k-approximations, i.e., when k is a large 2 1 2 2 3/2 dz [ 2 (k z vi + vi vn)] n i=1 k vn n number (k >> 0), ǫi meets the conditions of the binomial − X 2 approximation and is such that ǫ ǫ for all i, j . As d f i j By inspecting the numerator of each term in 2 , every ≈ ∈M dzn such, let ǫi = ǫ for all i , φ can be approximated as: 2 numerator can be reduced to (vi vn)(k vn) which is ∈M 2 n−2 d f − positive. Hence, 2 is a summation of positive terms (at √vi √vn−1 dzn φ 1+ . (29) least one of which is strictly positive since at least one ≈ √v (1 + ǫ/2) − √v (1 + ǫ/2) 2 i=1 n n d f X vi > vn for some i ). Hence, 2 > 0 and fk(zn) is ∈ M dzn Hence, since vi > vn−1 for i 0 min − ⇒ strictly convex. To obtain the sign of zn which minimizes ξ < 0 H1 is not positive definite H is not negative min ⇒ ⇔ fk(zn), we inspect the first derivative of fk(zn). Since zn definite; for any practically large approximation index k. is a minimum of f (z ), we must have: ∗ k n Case 3. zn−1 > 0, and zn < 0: We express ξ as ξ = n−1 vi min vn ∗ z z − Ξ, where df vn n vn−1 n 1 =1+ =0. min 2 dzn zn=zn 1 2 min n−1 i=1 2 k z vi + vi vn k vn n vi zn X − Ξ=1+ . (30) q (35) v 2 2 i=1 n znvi/vn + (vi vn)/(k vn) X − Since the denominator of the summation term in (35) is Since z∗ > 0, thep sign of ξ is the same as the sign of n−1 always positive, and since vi/vn for all i is always Ξ. By taking the first derivative of Ξ with respect to k, we positive, zmin must be negative to solve (35).∈ M ∂Ξ ∗ n can see that ∂k < 0, since zn < 0 and zi > 0 for all i and v > v for all i . Hence Ξ is decreasing∈M in k. Proposition 2 is valuable for characterizing the possible i n ∈ M Knowing that k> 0, the upper bound of Ξ, Ξ¯, is then solutions in zn to fk(zn)= D, and equivalently to (14), as min stated in Corollary 1. To this end, let D , fk(z ) be the n−1 n v z minimum value of f (z ) which is unique given that f (z ) Ξ¯ = lim Ξ=1+ k i n (31) k n k n k→0 v (v v ) · was proven to be strictly convex. i=1 n i − n X Corollary 1. f (z )= D has: a) two solutions if D>D, Hence, ξ is negative when p k n b) one solution if D = D, or c) no solution if D − − (32) Proof. From Proposition 2, we know that fk(zn) is a strictly n 1 vizn · i=1 √vn(vi−vn) convex function. Therefore any line D parallel to the zn axis P will intersect fk(zn) in two points if D>D, one point if Here, we note that vn has the same order of magnitude as D = D, and will not intersect fk(zn) if D> 0,− the condition in (32) always holds true. Thus, this Based on Corollary 1 and Theorem 1, we can conclude that p implies that H1 is not positive definite and, hence, H is not for D D, we have one or two possible local maxima for negative definite for any practically large k. U˜ a(z,≥ k). However, we must check whether these maxima ∗ ∗ Case 4. zn−1 > 0, and zn > 0: For this case, ξ > 0, are feasible as they should correspond to zn > 0, which is a hence, H1 is positive definite and as a result H is negative necessary condition as proven in Theorem 1. These maxima ∗ definite. Hence, only for zn−1 > 0, and zn > 0, H is also must satisfy the following feasibility conditions on zi positive definite. for i : ∈M Let us define the solutions of L+(zn)= D and L−(zn)= 6 D as z¯n and zn, respectively; then we have: 5 ∗ min ∗ z¯n > z¯n zn zn >zn. (40) 4 ≥ ≥ These inequalities are illustrated in Fig. 2. 3 min ∗ From Proposition 2, we have zn < 0, and, thus, zn < 0. 2 ∗ Hence, zn < 0 cannot be an acceptable solution for f(zn)= 1 D since it violates the condition in (37). We next investigate ∗ ∗ a z¯n. For it to be a valid solution, we must have 0 < z¯n < R 0 ∗ -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 and the resulting zi , which can be computed from (15), must satisfy the constraints in (37). Since z¯n is an upper bound of Fig. 2: Comparison of solutions of equations fk(zn) = ∗ z¯n, we start by studying z¯n. In this respect, next, we prove D,L+(zn)= D, and L−(zn)= D. a that z¯n < R : a a b b b a b a − ri R , ri R , R zi = ri ri R . (36) n 1 ≤ ≤ ⇒− ≤ − ≤ vi L+(zn)= D, zn 1+ = D, Therefore, for any solution z∗ > 0 of f(z ) = D, we ⇒ v n n i=1 r n ! must check that z∗, for i , satisfy X i ∈ N D ∗ z¯n = . (41) a n−1 0 1 and 0 0, we must have fk(0) D, which yields: P ≤ n−1 Proof. From Corollary 1, we know that, for D > D, vi vn ∗ ∗ − Dk. (42) fk(zn)= D has two solutions which we denote as zn < z¯n. v ≤ i=1 r n We need to prove that only one of these solutions satisfies X (37). First, we prove that fk(zn) has two slant asymptotes Under the condition on D in (42), we obtain 0 < ∗ a ∗ ∗ L1(zn) and L2(zn), where fk(zn) > L+(zn),L−(zn). To z¯n < R . However, for zn =z ¯n to be a valid solution, ∗ prove this statement, we find the linear approximations of the resulting zi , for all i which can be computed ∈ M∗ a fk(zn) as zn goes to + and , as follows: from (15), must satisfy 0 < zi R . From Theorem 1, ∞ −∞ n−1 ∗ ≤ ∗ a we know that zi > 0. To prove that zi R for all vi ∗ ≤ L+(zn) = lim fk(zn)= zn + zn i , the maximum zi for all i must be less zn→∞ vn ∈ M ∈ M i=1 r that Ra. Consider an index i that corresponds to such − X n 1 ∗ ∗ 1 2 ∗2 vi maximum zi . Since zi = k2v (k z¯n vi + vi vn) > = zn 1+ , (38) n − vn 1 2 2 ∗ i=1 ! 2 (k z¯ v + v v ) (dueq to z¯ < z¯ ), we show that r k vn n i i n n n X n−1 − vi 1 2 2 a q 2 (k z¯ vi + vi vn) R . Therefore, we must have: L−(zn) = lim fk(zn)= zn + ( zn) k vn n zn→−∞ − v − ≤ i=1 n X r q n−1 2 vi vi vn a vi z¯n + − 2 R , = z 1 . (39) vn vnk ≤ n − v r i=1 n ! D2 v v v X r i i n a2 2 + − R , ⇔ n−1 v v k2 ≤ Moreover, to prove that fk(zn) >L+(zn) we must have: 1+ vi n n i=1 vn − − n 1 n 1 q 2 1 2 2 vi P D vi vi vn a2 zn + (k z vi + vi vn) >zn 1+ + R . (43) 2 n n 2 − 2 k vn − vn ⇔ vnk ≤ i=1 r i=1 r ! i=1 √vi X n−1 n−1 X − vi vi vn vi b n 1 a a Since D< (n P1)R = n R R > (nD)/(n 1), zn + 2−2 >zn , − ⇒ − ⇔ vn k z vn vn and the maximum value for v is 1, then, by comparing the i=1 r n i=1 r i X X maximum value of the left-hand side of the inequality in (43) 2 2 which always holds true since (vi vn)/(k znvn) 0 for with the minimum value of the right-hand side, we can find i ; with at least one strict inequality− for an i≥ . ∈ M ∈ M a minimum value for k as follows: Hence, fk(zn) > L+(zn). Using a similar approach, we 2 1 2 n 2 can also prove that − . From fk(zn) > L (zn) fk(zn) > D + 2 D ( ) , vnk ≤ n 1 L (z ),L−(z ), we can state that f (z ) always lies in the + n n k n n −1 upper subspace of the intersection of its asymptotes as shown Dk − . (44) ≥ v (2n 1) in Fig. 2. n − p From (42) and (44), we can conclude the proof by Proof. As stated in (10), the pure-strategy NE of the GCBG stating that the unique maximum of U˜ a(z, k), for Dk is the set of two vectors ra∗ and rb∗, which solve: ≥ n−1 n−1 vi−vn ∗ ∗ max , i=1 v , is z such that zi for ˜ a a b √vn(2n−1) n min max U (r , r , k). (49) rb∈Qb ra∈Qa i is as defined in (15),q and z∗ =z ¯∗. ∈M P n n ˜ a a b a In Theorem 2, we derived a unique local maximum of Maximizing U (r , r , k), with respect to r is similar ˜ a − to maximizing U (z, k) with respect to z, where z = ˜ a n−1 n 1 vi−vn U (z, k) for Dk max , i=1 v . In a b ≥ √vn(2n−1) n r r . In Theorem 3, we have shown that, if Dk − − ≥ Theorem 3, we prove that this local maximumq is actually a n−1 n 1 vi−vn ˜ a P max , i=1 v , then U (z, k) has only global maximum of U˜ a(z, k). √vn(2n−1) n ∗ ∗ ∗ ∗ ∗ one maximim at zP= [zq1 ,...,zn−1, z¯n], where z¯n is the Theorem 3. The unique local maximum z∗ = ∗ positive solution of fk(zn) = D, and zi is as defined in ∗ ∗ ∗ ∗ ∗ ∗ ∗ [z1 ,...,zn−1,zn], where zn =z ¯n and zi = (15). Therefore, ra can be written as follows: 1 2 2 2 (k znvi + vi vn) for i , is a k vn − ∈ M ra∗ = rb + z∗. (50) uniqueq global maximum of U˜ a(z, k) for Dk − ≥ ∗ n−1 n 1 vi−vn By substituting ra in U˜ a(ra, rb, k), we will have: max , i=1 v . √vn(2n−1) n n P q v v Proof. Theorem 2 showed that the derived local maximum U˜ a(ra∗, rb, k)= i arctan k(ra∗ rb) + i of ˜ a z for a given limit on is unique. Next, we π i − i 2 U ( , k) Dk i=1 a prove that U˜ (z, k) has no local minimum. Xn v 1 The proof follows the same logic as that of Theorem 1 by = i arctan(k(z∗)) + , (51) π i 2 attempting to find solutions in (18) which lead to a positive i=1 X definite Hessian matrix H. In this regard, H is positive which has a constant value. Therefore, the minimizing vector definite H1 is negative definite all pivots of H are 2 rb∗ consists of all the strategies in player b’s strategy set, b. negative.⇔ As such, a local minimum⇔ must have all ∗ zi < 0 Therefore, the pure strategy NE of the GCB is as definedQ in for obtained from (18) and must have i n 1 ξ < 0 (45)-(48). Moreover, the value of the game for each player which∈ is M\{ defined− in} (27). To this end, similarly to the four- can be expressed as follows: step case analysis derived in Theorem 1 over the signs of zn n ∗ ∗ 1 vi ∗ and zn−1, we can prove that for zi < 0 for i n 1 V a(k) , 1 V b(k)= + arctan(kz ) . (52) ∗ ∈M\{ − } − 2 π i no feasible zn and zn−1 lead to ξ < 0. In fact, it can be i=1 derived that (z∗ < 0,z < 0) and (z∗ > 0,z > 0) X n−1 n n−1 n lead to non-feasible solutions that violate (14) while (z∗ > n−1 From Theorem 4, we can see that the GCB admits an 0,z < 0) and (z∗ < 0,z > 0) lead to ξ > 0, and hence, n n−1 n infinite number of pure-strategy NEs. However, we have a non-positive definite H. characterized all of them and we have also shown that, by ∗ ∗ ∗ 1 2 2 Therefore, z =z ¯ and z = 2 (k z vi + vi vn) n n i k vn n − deviating from one NE to the other, the payoff of each player for i is a unique local maximumq of U˜ a(z, k) and does not change. ∈ M U˜ a(z, k) admits no local minima. Hence, this unique local Moreover, Theorem 4 shows that the player with the higher maximum is a global maximum. available resources can always achieve a higher payoff by playing the derived pure-strategy NE, since z∗ > 0 for Based on the derived maximum of U˜ a(z, k) with respect i which yields a b . This implies that, if to z, we next characterize the pure-strategy NE for the i V (k) > V (k) both∈ N players are fully rational, the more resourceful player GCBG. wins the game in aggregate, which captures a missing feature − 1 n−1 1 n 1 vi−vn in the CBG. Indeed, in the CBG, even when a player is Theorem 4.For k max D , D i=1 v , ≥ √vn(2n−1) n more resourceful, it cannot find a winning pure strategy. In the pure-strategy NE for the GCB game is theq allocation P this respect, based on the k approximation in the proposed vectors defined as follows: GCBG, we can significantly− increase the approximation b∗ b∗ b ∗ r = [r1 ,...,rn ], (45) index k so that the GCBG very closely approximates the ∗ ∗ CBG and still be able to derive a solution to the game in ra = rb + [z∗,..., z¯∗], (46) 1 n pure strategies. Here, we note that, when k goes to infinity, ∗ ∗ where zn =z ¯n (the positive solution of fk(zn)= D), and our derived solution will no longer hold since the first order conditions in (17) will be met for any z (i ). This is v v v i ∈ M z∗ = z∗2 i + i − n , for i (47) aligned with the CBG which does not admit an NE in pure i n v v k2 ∈M r n n strategies, and indeed, confirms that when k goes to infinity and our proposed GCBG converges to the classical CBG. Next, ∗ ∗ we illustrate the GCBG and derived pure-strategy NE using rb + + rb = Rb. (48) 1 ··· n an example. 1 1

0.9 0.8 0.8

0.7 0.6

0.6 0.4 0.5

0.4 0.2

0.3 0 0.2 1 2 3 4 5 6 7 8 9 10

0.1 Fig. 5: Payoffs of the players at NE with D =5. 1 2 3 4 5 6 7 8 9 10 to practical applications. For this new class of CBG, we have Fig. 3: Difference between the allocation of resources at NE proven the existence of a pure-strategy NE and provided 1 a detailed derivation of closed-form analytical expressions

0.8 of the NE. Finally, we have presented a numerical example which illustrates the effects of the increase in the number 0.6 of resources of each player on the game’s outcome and NE strategies. Our results also showcased the effect of the 0.4 approximation index on the solution of the game. 0.2 REFERENCES 0 [1] Y. Wu, B. Wang, K. J. R. Liu, and T. C. Clancy, “Anti-jamming games -10 -8 -6 -4 -2 0 2 4 6 8 10 in multi-channel cognitive radio networks,” IEEE Journal on Selected Areas in Communications, vol. 30, no. 1, pp. 4–15, January 2012. [2] W. Saad, Z. Han, M. Debbah, and A. Hjorungnes, “A distributed Fig. 4: Payoffs of the players at NE. coalition formation framework for fair user cooperation in wireless networks,” IEEE Transactions on Wireless Communications, vol. 8, IV. AN ILLUSTRATIVE EXAMPLE no. 9, pp. 4580–4593, September 2009. [3] A. Ferdowsi, A. Sanjab, W. Saad, and N. B. 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