Advances in Systems Science and Application (2015) Vol.15 No.1 45-59 System Compatibility: Price of Anarchy and Control Mechanisms in the Models of Concordance of Private and Public Interests Olga I. Gorbaneva and Guennady A. Ougolnitsky Southern Federal University, Russia Abstract The problem of system compatibility is considered. Its solution ensures maximiza- tion of the social welfare by consideration of individual interests of the agents. The conditions of system compatibility and respective control mechanisms are analyzed for the models of concordance of common and private interests in the agents’ resource allocation. Keywords Concordance of interests; Control mechanisms; Hierarchical game the- ory; System compatibility 1 Introduction A problem of concordance of interests in the active systems may be considered in two aspects. First, it is well known that an egoistic behavior of independent active agents often implies a less social welfare than in the case of their coor- dinated actions. The quantitative side of this problem is named “inefficiency of equilibria”[1] and can be characterized by the price of anarchy index introduced by Papadimitriou [2].Second, the agents can allocate their resources between pri- vate and public interests. In the seminal paper by Germeier and Vatel [3],it is shown that if payoff functions of all agents are convolutions by minimum of the functions of public and private interests then in the respective game there is a Pareto-optimal Nash equilibrium (i.e. the price of anarchy is equal to the ideal value of one). We continue to investigate models of that type in literature [4,5] and in the present paper. Mathematical methods of solution of the static problems of concordance of interests of active agents are developed in the theory of incentives [6], the infor- mation theory of hierarchical systems [7,8], the theory of control in organizations [9,10], mechanism design [1]. It should be noticed that the mechanism design investigates another setting of the problem: how to motivate agents to report the true information about their type (the problem of strategy-proofness, or incentive compatibility). In the present paper a notion of system compatibility is introduced. The sys- tem compatibility means that individually optimal controls of agents form the globally optimal vector of controls for a social welfare function. This setting is close to the problem of meta-game synthesis in the theory of active systems [11]. Conditions of the system compatibility are studied for the models of concordance of private and public interests (CPPI-models). As the conditions are quite re- 46 Olga I. Gorbaneva and Guennady A. Ougolnitsky: System Compatibility,Price of Anarchy and... strictive, the control mechanisms directed to ensure the system compatibility are proposed. In the framework of the authors’ concept [12], the administrative and economic mechanisms with/without a feedback are constructed. The rest of the paper is organized as follows. In the section 2 the principal notions such as price of anarchy, system compatibility, control mechanisms, and CPPI-models are introduced. Economic mechanisms without feedback and with it as well as algorithms of their implementation are considered in the sections 3 and 4 respectively. Administrative control mechanisms are discussed in the section 5. Section 6 concludes. 2 Principal Notions. Let’s consider a set N = {1, 2, ..., n} of active agents maximizing their payoff functions gi(u1, ..., un) → max (1) s.t. ui ∈ Ui, i ∈ N (2) Let the solution of the game (1)-(2) be a Nash equilibrium uNE ∈ NE,u = (u1, ..., un). Introduce a utilitarian social welfare function. g0(u) = ∑ j∈N gj(u). Let umax be a solution of the maximization problem g0(u) → max, u ∈ U = U1 × ...× Un (3) Denote gmax 0 = g0(u max) . Definition 1. The model(1)-(3)is system compatible if ∀uNE ∈ NE g0(u NE) = gmax 0 . A quantitative measure of the system compatibility is the price of anarchy [1-2] PA = min uNE∈NE g0(u NE) gmax 0 (4) It is evident that a model is system compatible if PA = 1 . The system com- patibility is a rare phenomenon, and it is worthwhile to use control mechanisms for its achievement[9]. Suppose that maximization of the social welfare (3) is the objective of a specific agent (Center, principal, mechanism designer and so on) who has an ability of impact on the sets of feasible controls and/or payoff functions of the other agents to provide this objective. Denote the first possibility as Ui = Ui(qi) , and the second one gi = gi(pi, ui), where q, p are vectors of the principal’s administrative and economic controls respectively. In the context of literature[12] we can differ- entiate the following control mechanisms (methods of control). Advances in Systems Science and Application (2015) Vol.15 No.1 47 Table 1 Control mechanisms Principal’s impact Without a feed- back (Γ1) With a feedback (Γ2) On the sets of feasible controls of the agents(administrative one,or com- pulsion) qi = const qi = qi(u) On the agents’ payoff function- s(economic one,or impulsion) pi = const pi = pi(u) Thus, the principal can exert influence on the sets of feasible controls of the agents (administrative control mechanism, or compulsion) or on the agents’ pay- off functions (economic control mechanism, or impulsion). Both mechanisms can include not or include a feedback on control. In the first case a hierarchical game of the type Γ1(Stackelberg game) holds, meanwhile the second case generates a hierarchical game of the type Γ2(Germeier game). So, four types of control mechanisms are possible (Table 1). Notice that now or in dependence of the used control mechanism. Definition 2. A control mechanism q(p) in the model (1)-(3) is system compat- ible if gmax 0 = g0(u NE(q)) or gmax 0 = g0(u NE(p)) respectively. For definiteness let’s specify the model (1)-(3) in the form gi(u) = pi(ri − ui) + sic(u) → max, 0 ≤ ui ≤ ri, i ∈ N (5) g0(u) = ∑ j∈N pj(rj − uj) + c(u) → max, ∑ j∈N sj = { 1, ∃i : si > 0, 0, otherwise. (6) Here, ri > 0 is a resource of the i-th agent; ui is a part of the resource assigned for production of the public payoff described by a function c(u); si is a share of the i-th agent in the public payoff; pi(ri − ui) is a function of the i-th agent’s private interest. The functions pi, c are supposed to be continuously differentiable and concave in all arguments. So, each agent shares his resource between public and private interests according to the ratio ui and ri − ui respectively. Thus, the model (5)-(6) describes a concordance of the private and public interests in resource allocation. Our investigation of models of the type (5)-(6) (CPPI- models) develops the approach by Germeier and Vatel and Burkov and Opoitsev [3,11]. Economic control mechanisms in the model (5)-(6) are implemented by the principal’s choice of the values si. To use administrative mechanisms one should suppose additionally that the principal can bound feasible controls of the agents: q̃i ≤ ui ≤ q̄i, i ∈ N (7) 48 Olga I. Gorbaneva and Guennady A. Ougolnitsky: System Compatibility,Price of Anarchy and... The control mechanisms from the Table 1 can be specified for the CPPI-model (5)-(7) as follows (Table 2). Table 2 Mechanisms of system compatibility in CPPI-models Principal’s impact Without a feedback (Γ1) With a feedback (Γ2) On the sets of fea- sible controls of the agents(administrative one,or compulsion) q̃i ≤ ui ≤ q̄i, q̃i, q̄i = const q̃i(u) ≤ ui ≤ q̄i(u) On the agents’ pay- off functions(economic one,or impulsion) si = const si = si(u) 3 Economic Control Mechanisms Without a Feedback. Assume that in the model (5)-(6) ∀i si = const .Using the first order conditions we get that the internal system compatibility in the model (5)-(6) holds only if ∂c ∂ui = 0, i ∈ N (8) Let’s notice that it is true also for models with partly coincident interests in more general form [5] gi(u) = pi(ui) + sic(u) → max, ui ∈ Ui, i ∈ N (9) So, the following statement is true. Theorem 1. Suppose that ∃i ∈ N : ∂c/∂ui ̸= 0. Then for system compatibility in the model (5)-(6) it is necessary that ∀i ∈ N ui = 0 ∨ ui = ri . In other words, the system compatibility in the model (5)-(6) is possible only if all agents are pure individualists (ui = 0) or pure collectivists (ui = ri) . Example 1 (linear CPPI-model). Consider a linear specification of the model (5)-(6): gi(u) = ki(ri − ui) + siK ∑ j∈N uj → max, 0 ≤ ui ≤ ri, i ∈ N (10) g0(u) = ∑ j∈N kj(rj − uj) +K ∑ j∈N uj → max; S : 0 ≤ si ≤ 1, ∑ j∈N sj = { 1, ∃i : s > 0, 0, otherwise. (11) Advances in Systems Science and Application (2015) Vol.15 No.1 49 Here ki > 0,K > 0 are given constants which characterize efficiencies of the functions of private and public payoffs respectively. The first order conditions for the agents and the principal give respectively: ∂gi ∂ui = Ksi − ki { ≥ 0, si ≥ ki K ⇒ uNE i = ri; < 0, si < ki K ⇒ uNE i = 0; ∂g0 ∂ui = K − ki { ≥ 0, ki ≤ K ⇒ umax i = ri; < 0, ki > K ⇒ umax i = 0. Thus, the system compatibility is possible and holds on the bounds of the segments of feasible controls. Two partitions of the set N can be defined: N = I0 ∪ C0, I0 ∩ C0 = ∅ : I0 = {i ∈ N : ki > K ⇒ umax i = 0}(immanent individualists); C0 = {i ∈ N : ki ≤ K ⇒ umax i = ri}(immanent collectivists). N = I(s) ∪ C(s), I(s) ∩ C(s) = ∅ : I(s) = {i ∈ N : si < ki K ⇒ uNE i = 0}(controlled individualists); C(s) = {i ∈ N : si ≥ ki K ⇒ uNE i = ri}(controlled collectivists). The immanent partition is determined by the objective properties of the model (10)-(11), meanwhile the controlled partition results from the optimal reaction of the agents on the choice of a vector s = (s1, ..., sn) by the principal. Notice that i ∈ I0 ⇒ ki/K > 1 ⇒ ∀s ∈ S si ≤ ki/K ⇒ i ∈ I(s), i.e. ∀s ∈ S I0 ⊂ I(s) . The inverse statement is wrong: let ki ≤ K ⇒ i ∈ C0 but si = 0 ⇒ i ∈ I(s). Similarly, it is simple to show that ∀s ∈ S C(s) ⊂ C0. It is also clear that if ∃s ∈ S : I(s) = I0, C(s) = C0 then the model is system compatible. To find her optimal control s∗ the principal should solve a discrete optimization problem g0(s) = ∑ j∈I(s) kjrj +K ∑ j∈C(s) rj → max (12) S : 0 ≤ si ≤ 1, i ∈ N, ∑ j∈N sj = { 1, ∃i : si > 0, 0, otherwise. (13) if C0 = ∅ then the model is system compatible for ∀s ∈ S .In this case N = I(s) = I0, g max 0 = gI0 = ∑ j∈N kjrj (individualistic society). 50 Olga I. Gorbaneva and Guennady A. Ougolnitsky: System Compatibility,Price of Anarchy and... Suppose that C0 ̸= ∅ if ∑ i∈C0 ki ≤ K then let for i ∈ C0, s ∗ i = ki K + εi so that∑ i∈C0 s∗i = 1. Then (s∗) = C0 , and the model is system compatible. Particularly, if I0 = ∅ then N = C0 = C(s∗), gmax 0 = gC0 = K ∑ j∈C0 rj (collectivistic society). In general case I0 ̸= ∅, C0 ̸= ∅, ∑ j∈C0 kj > K. Here |C0| > 1. Subject to ∀s ∈ S , I0 ⊂ I(s) it is impossible to put any i ∈ I0 into C(s), thus the problem (12)-(13) is reduced to the construction of a set C(s) ⊂ C0 such that g0(s) = ∑ j∈I(s)=C0\C(s) kjrj +K ∑ j∈C(s) rj → max, ∑ j∈C(s) kj ≤ K, s ∈ S This problem is still being solved. Example 2 (power-linear CPPI-model). Consider the following specification of (5)-(6): gi(u) = ki √ ri − ui + siK ∑ j∈N uj → max, 0 ≤ ui ≤ ri, i ∈ N (14) g0(u) = ∑ j∈N kj √ rj − uj +K ∑ j∈N uj → max, 0 ≤ si ≤ 1, ∑ j∈N sj = { 1,∃i : s > 0, 0, otherwise. (15) The first order conditions for the agents and the principal give respectively: 0 = ∂gi ∂ui = Ksi − ki 2 √ ri − ui ⇒ uNE i = { ri − k2i 4K2s2i , si ≥ ki 2K √ ri ; 0, otherwise; 0 = ∂g0 ∂ui = K − ki 2 √ ri − ui ⇒ umax i = { ri − ki 4K2 , ki ≤ 2K √ ri; 0, otherwise. Thus, the system compatibility in the model (14)-(15) holds only if ∀i ∈ N , ki ≥ 2K √ ri (16) If this condition holds, then N = I(s) = I0, gmax 0 = gI0 = ∑ j∈N kj √ rj for any s ∈ S. If (16) doesn’t hold, then the problem of system compatibility can be formu- lated in a weaker form of maximization the price of anarchy (4) by a mechanism of economic control. The following problem of discrete optimization arises g0(s) = ∑ j∈I(s) kj √ rj + ∑ j∈C′(s) [Krj + kj 2 2Ksj − k2j 4Ks2j ] → max (17) Advances in Systems Science and Application (2015) Vol.15 No.1 51 s.t. (13), where I(s) = {i ∈ N : si ≤ ki 2K √ ri ⇒ uNE i = 0}, C ′(s) = {i ∈ N : si > ki 2K √ ri ⇒ uNE i = ri − k2i 4K2s2i } This problem is reduced to the following one: ∑ j∈C′(s) ( 2k2j sj − k2j s2j ) + λ( ∑ j∈C′(s) sj − 1) → max (18) The FOC gives: −2k2j 2s2j + k2j 2s3j + λ = 0, s.t. s ∈ S. Multiplication by sj gives the system s3j − 2k2j sj λ + 2k2j λ = 0∑ i∈C′(s) si = 1 To find sj and λ the following should be done: 1) to express sj by λ from the first equation; 2) to substitute the expression into the second equation and find λ; 3) to substitute the value of λ back to the expression from 1) and find the respective si. Given λ the first equation can be solved analytically by Cartan method (the solution is omitted due to its awkwardness) or numerically. As only real solutions such as 0 ≤ si ≤ 1 are feasible, the following conclusions can be received: (1) a solution exists only if λ < 0. Rewriting the equation as s3j = 2k2j λ (sj − 1), we get λ < 0 due to positive left part and negative terms in the numerator and in the brackets; (2) when λ < 0 the only real solution si of the equation exists, and 0 ≤ si ≤ 1. Actually, lets find the derivative of the expression f (sj) = s3j − 2k2j sj λ + 2k2j λ : f ′ (sj) = 3s2j − 2k2j λ > 0. Therefore, the function f(si) increases in R , and if a root of the equation f(si) = 0 exists then it is unique.Now let’s prove that the root exists and satisfies 0 ≤ si ≤ 1. f (0) = 2k2j λ < 0, f (1) = 1 > 0. Subject to continuousness of f(si) the property is proved. 52 Olga I. Gorbaneva and Guennady A. Ougolnitsky: System Compatibility,Price of Anarchy and... (3) sj increases in λ. Let’s find the derivative ∂sj ∂λ of an implicit function: 3s2j · s′j − 2k2jλ·s′j−2k2j sj λ2 − 2k2j λ2 = 0, or s′j = 2k2j (1−sj) λ(3s2jλ−2kj) < 0 that implies. (4) ∑ i∈C′ si increases in λ. Therefore, it is possible to choose such λ that ∑ i∈C′(s) si = 1. Thus, a problem with two variables is reduced to a problem with one variable. The problem can be solved by the method of dichotomy in which the left bound of a segment is equal to λ1 : ∑ i∈C′(s) si > 1 (for example,λ1 = −ε √ b2 − 4ac , and the right bound is a big enough λ2 : ∑ i∈C′(s) si < 1. 4 Economic Control Mechanisms with a Feedback. Now assume that in the model (5)-(6) si = si(ui) or even si = si(u). According to FOC, the internal system compatibility is possible only if ∂si(u) ∂ui c(u) = [1− si(u)] ∂c(u) ∂ui , i ∈ N (19) This condition is less restrictive than (8) when si = const. Both empirical and theoretical approaches can be used in the following analysis. In the context of empirical approach widely spread in practical activity methods of resource allo- cated are investigated. For example, consider a natural method of proportional allocation si(u) =  ui∑ j∈N uj , ∃m : um > 0, 0, otherwise. (20) In this case (19) takes the form ∑ j ̸=i uj [ ∂c(u) ∂ui ∑ j∈N uj − c(u)] = 0, i ∈ N and therefore the following statement is evident. Theorem 2. The mechanism of proportional allocation (20) is system com- patible in the CPPI-models with linear function of public payoff c(u) and any functions of private payoffs. Example 3. Suppose that gi(u) = ki √ ri − ui+siK ∑ j∈N uj , where is determined Advances in Systems Science and Application (2015) Vol.15 No.1 53 by (20). Then ∂gi ∂ui = ∂g0 ∂ui = K− ki 2 √ ri−ui , uNE i = umax i = { ri − k2i 4K2 , ki ≤ 2K √ ri, 0, otherwise; gmax 0 = ∑ j∈I kj √ rj+K ∑ j∈C′ rjI = {i ∈ N : ki > 2K √ ri}, C ′ = {i ∈ N : ki ≤ 2K √ ri}. Theoretical approach to building of system compatible economic impulsion mechanisms is based on Germeier’s theorem for games of the type Γ2 (see Ap- pendix). Let’s apply this theorem to the linear model (5)-(6). We obtain sDi is arbitrary (g0 does not depend on s ), sPi ≡ 0; Li = kiri; Ei = {ui = 0}; Di = {(si, ui) : si > kiui K ∑ j∈N uj , n∑ i=1 si = 1}; K2 = gI0 = ∑ j∈N kjrj To find K1 we must solve an optimization problem g0(u) = ∑ j∈I(s) kj(rj − uj) +K ∑ j∈C(s) uj → max s.t. kiui K ∑ j∈N uj < si ≤ 1, ∑ j∈N sj = { 1, ∃i : si > 0, 0, otherwise, , 0 ≤ ui ≤ ri, i ∈ N From the first order condition ∂g0 ∂ui = K − ki ⇒ u∗i = { ri, ki ≤ K (set C0), 0, otherwise (set I0). It is proved that it is possible to find si from the set Di: si∈C0 = kiri K ∑ j /∈I0 rj + εi, n∑ i=1 εi = 1− ∑ i/∈I0 kiri K ∑ j /∈I0 rj ; si∈I0 = 0. Therefore, like in Γ1 formulation (example 1), we obtain g0(u ∗) = ∑ j∈I0 kjrj +K ∑ j∈C0 rj 54 Olga I. Gorbaneva and Guennady A. Ougolnitsky: System Compatibility,Price of Anarchy and... Three cases are possible: 1. ∀i ∈ N , ki > K ⇒ u∗i = 0, K1 = gmax 0 = gI0 = ∑ j∈N kjrj , in this case K2 = K1, and gi = kiri = Li . 2. ∀i ∈ N , ki ≤ K ⇒ u∗i = ri, K1 = gmax 0 = gC0 = K ∑ j∈N rj , that is K2 ≥ K1, and gi = Kri > Li, hence, condition si > kiui K ∑ j∈N uj is satisfied. In this case N = C0 = C(s), I = ∅, gmax 0 = gC0 = K ∑ j∈N rj , s∗, is any allocation satisfying (13) (“collectivistic” society). 3. ∃l,m : kl > K, km ≤ K. Here the solution of the problem (12)-(13) is given by the following algorithm: to assign s∗i =  kiri K ∑ j /∈I0 rj + εi, ki ≤ K; 0, ki > K. Then u∗i = { ri, ki ≤ K, 0, ki > K, and gmax 0 = K ∑ j∈I0 rj + ∑ j /∈I0 kjrj . In this case C0 = C(s) ̸= ∅, I0 = I(s) ̸= ∅ (≪mixed≪ society). In all cases 1-3 there is a system compatibility of the model (12)-(13) and economic impulsion mechanism s∗. gi(u) = ki √ ri − ui + siK ∑ j∈N uj → max, 0 ≤ ui ≤ ri, i ∈ N g0(u) = ∑ j∈N kj √ rj − uj +K ∑ j∈N uj → max, 0 ≤ si ≤ 1, ∑ j∈N sj = { 1,∃i : s > 0, 0, otherwise. We obtain sDi is arbitrary (g0 does not depend on s ),sPi ≡ 0; Li = ki √ ri; Ei = {ui = 0}; Di = {(si, ui) : si > kiui K( √ ri + √ ri − ui) ∑ j∈N uj }; K2 = gI0 = ∑ j∈N kj √ rj . To find K1 we must solve an optimization problem g0(u) = ∑ j∈N kj √ rj − uj +K ∑ j∈N uj → max ki( √ ri − √ ri − ui) K ∑ j∈N uj < si ≤ 1, ∑ j∈N sj = { 1, ∃i : si > 0, 0, otherwise, , 0 ≤ ui ≤ ri, i ∈ N Advances in Systems Science and Application (2015) Vol.15 No.1 55 From the first order condition u∗i = { ri − ki 2 4K2 , ri ≤ 2K √ ri; 0, otherwise. It is proved that it is possible to find si from the set Di: s∗i =  ki( √ ri− ki 2K ) K ∑ j /∈I0 ( rj− kj 2 4K2 ) + εi, ui > 0; 0, ui = 0. where n∑ i=1 εi = 1− ∑ i/∈I0 ki( √ ri− ki 2K ) K ∑ j /∈I0 ( rj− kj 2 4K2 ) ; Therefore K1 = ∑ j∈I0 kj √ rj + ∑ j /∈I0 ( Kri + k2i 4K ) > K2. The following cases are possible. 1. ∀i ∈ N , ki > 2K √ ri ⇒ u∗i = 0, K1 = gmax 0 = gI0 = ∑ j∈N kj √ rj , in this case K2 = K1, and gi = ki √ ri = Li. 2. ∀i ∈ N , ki ≤ 2K √ ri ⇒ u∗i = ri, K1 = gmax 0 = ∑ j∈N ( Kri + k2i 4K ) , i.e. K2 ≤ K1, Li = ki √ ri; Ei = {ui = 0}; Di = { (si, ui) : si > kiui K( √ ri+ √ ri−ui) ∑ j∈N uj } ; and gi = Kri + k2i 4K > Li, hence, condition si > kiui K( √ ri+ √ ri−ui) ∑ j∈N uj is satisfied. In this case N = C ′, I0 = ∅, ∑ j∈N ( Kri + k2i 4K ) , s∗ is any allocation satisfying (13) (“collectivistic” society). 3. ∃l,m : kl > 2K √ rl, km ≤ ki > 2K √ rm. Here the solution of the problem (12)-(13) is given by the following algorithm: to assign s∗i =  ki( √ ri− ki 2K ) K ∑ j /∈I0 ( rj− kj 2 4K2 ) i , ki ≤ 2K √ ri; 0, ki > K. Then u∗i = { ri − ki 2 4K2 , ri ≤ 2K √ ri; 0, otherwise. and gmax 0 = ∑ j∈I0 kj √ rj + ∑ j /∈I0 ( Kri + k2i 4K ) . In this case C0 = C(s) ̸= ∅, I0 = I(s) ̸= ∅ (≪mixed≫ society). In all cases 1-3 there is a system compatibility of the model (12)-(13) and economical impulsion mechanism s∗. 5 Administrative control mechanisms. Suppose that the principal can bound the agents’ sets of feasible controls. Con- sider the case of administrative control without a feedback. Then the model (5)-(6) takes the form gi(q̃i, q̄i, u) = pi(ri − ui) + sic(u) → max, q̃i ≤ ui ≤ q̄i, si ∈ [0, 1]; (21) 56 Olga I. Gorbaneva and Guennady A. Ougolnitsky: System Compatibility,Price of Anarchy and... g0(q̃, q̄, u) = ∑ j∈N pj(rj − uj) + c(u) → max, 0 ≤ q̃i ≤ q̄i ≤ ri, i ∈ N. (22) It is clear that if there are no restrictions then the problem (22) has a trivial solution q̃i = q̄i = umax i , i ∈ N . Therefore the principals control cost should be considered. Then (22) takes the form g0(q̃, q̄, u) = ∑ j∈N pj(rj − uj) + c(u)− C(q̃, q̄) → max, 0 ≤ q̃i ≤ q̄i ≤ ri, i ∈ N, Where C(q̃, q̄) is a continuously differentiable and convex in all arguments com- pulsion cost function. Let’s give a simple example. Example 4. Assume that gi(q̃i, q̄i, u) = ki(ri − ui) + siK ∑ j∈N uj → max, q̃i ≤ ui ≤ q̄i, si ∈ [0, 1]; g0(q̃, q̄, u) = ∑ j∈N kj(rj − uj) +K ∑ j∈N uj − ∑ j∈N (m̃j q̃j + m̄j q̄j) → max, 0 ≤ q̃i ≤ q̄i ≤ ri, i ∈ N where K, ki, m̃i, m̄i are known positive constants. The FOC give ∂gi ∂ui = siK − ki ⇒ uNE i = { ri, ki ≤ siK, 0, ki > siK; ∂g0 ∂ui = K − ki ⇒ umax i = { ri, ki ≤ K, 0, ki > K; ∂g0 ∂q̃i = −m̃i, ∂g0 ∂q̄i = −m̄i ⇒ q̃max i = q̄max i = 0 Notice that if ki > K then ki > siK, therefore umax i = uNE i = 0, and com- pulsion is not required (the model is system compatible). Otherwise two cases should be differentiated: (a) ki ≤ siK ⇒ umax i = uNE i = ri, and compulsion is not required again; (b) siK < ki ≤ K ⇒ uNE i = 0, umax i = ri . In this case the princi- pal’s payoffs with and without compulsion should be compared. Denote by M the set of agents for whom siK < ki ≤ K. If compulsion holds we have g0(r, 0, r) = K ∑ j∈M rj − ∑ j∈M m̃jrj , and if not then g0(0, 0, 0) = ∑ j∈M kjrj . Thus, compulsion is rational if ∑ j∈M (K − m̃j − kj)rj > 0 holds. Advances in Systems Science and Application (2015) Vol.15 No.1 57 6 Conclusion. In the present paper the problem of system compatibility was analyzed. Its solu- tion ensures maximization of the social welfare by considering individual interests of the agents. This setting is close to the problem of mechanism design but dis- tinct because in the latter problem it is required to motivate agents to report the true information about their types instead of the direct maximization of the social welfare. The conditions of system compatibility are quite restrictive, therefore to pro- vide them it is worthwhile to construct control mechanisms. In the framework of authors’ concept, the mechanisms are classified by two attributes: direction of impact (sets of feasible controls of the agents or their payoff functions) and presence or absence of feedback in the control system. The first attribute differen- tiates administrative and economic control mechanisms (methods of compulsion and impulsion respectively), while the second one leads to hierarchical games of the types Γ1 and Γ2. The conditions and mechanisms of system compatibility are analyzed in the class of models of concordance of the private and public interests in resource allocation (CPPI-models). Some preliminary results about system compatibility of the CPPI-models are obtained. Thus, system compatibility in CPPI-models when si = const is reachable only if all agents are pure individualists (all re- sources are assigned for private interests) or pure collectivists (all resources are assigned for public interest). The exact dichotomous partition is built by spe- cific algorithms of discrete optimization. Economic mechanisms with a feedback simplify the achievement of system compatibility. Administrative mechanisms of system compatibility are under development. The research perspectives include: - investigation of the system compatibility for more general classes of models; - considering of corruption (an additional feedback on bribe); - analysis of dynamic settings, including phase constraints (requirements of sustainable development), investigation of the conditions of time consistence. Appendix (Germeier theorem). Assume that payoff functions of both play- ers M1(x1, x2),M2(x1, x2) are continuous on compact sets of feasible control- s X1, X2. Introduce the punishment function xP1 (x2) such that M2(x P 1 , x2) = min x1∈X1 M2(x1, x2), and the dominant strategy of the player 1 xD1 (x2), which sat- isfies the condition M1(x D 1 , x2) = max x1∈X1 M(x1, x2). Introduce also the following values and sets: L2 = max x2∈X2 M2(x P 1 (x2), x2); E2 = {x2 ∈ X2 : M2(x P 1 (x2), x2) = L2}; D2 = {(x1, x2) ∈ X1 ×X2 : M2(x1, x2) > L2}; 58 Olga I. Gorbaneva and Guennady A. Ougolnitsky: System Compatibility,Price of Anarchy and... K1 = sup (x1,x2)∈D2 M1(x1, x2) ≤ M1(x ε 1, x ε 2) + ε (D2 = ∅ ⇒ K1 = −∞); K2 = min x2∈E2 max x1∈X1 M1(x1, x2). Then the guaranteed payoff of the player 1 (Leader) in the game Γ2 (in which the first player knows the choice of the second player) is equal to w1 = max(K1,K2), and the respective -optimal guaranteeing strategy has the form x̃ε1(x2) =  xε1, x2 = xε2, K1 > K2, xD1 (x2), x2 ∈ D2, K1 ≤ K2, xP1 (x2), otherwise, where xε1 and xε2 are described above. Acknowledgements The work is supported by the Russian Foundation for Basic Research, project # 15-01-00432. References [1] N. Nisan, T. Roughgarden, E. Tardos and V. Vazirani. (2007), Algorithmic Game Theory, Cambridge University Press. [2] Papadimitriou C.H. 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(1974), “Game-theory analysis of a two-level hierarchi- cal control system”, USSR Computational Mathematics and Mathematical Physics,Vol.14,No.5,pp.72-81. Advances in Systems Science and Application (2015) Vol.15 No.1 59 [8] Kukushkin N.S. (1994), “A condition for existence of nash equilibri- um in games with public and private objectives”, Games and Economic Behavior,Vol.7,pp.177-192. [9] D. Novikov. (2013), Mechanism Design and Management: Mathematical Methods for Smart Organizations, Nova Science Publishers. [10] Novikov D. (2013), Theory of Control in Organizations, Nova Science Pub- lishers. [11] Burkov V.N. and Opoitsev V.I. (1974), “Metagame approach to the control in hierarchical systems”, Automation and Remote Control, Vol 35, No.1, pp.93-103. [12] Ougolnitsky G. (2011), Sustainable Management, Nova Science Publishers. Corresponding author Olga I. Gorbaneva, Ph.D. can be contacted at: gorbaneva@mail.ru Guennady A. Ougolnitsky, Ph.D. can be contacted at:ougoln@gmail.com