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1Contributions to Game Theory and Management, XVI, 74—86
Dynamic Cournot Oligopoly Models of the State Promotion of Innovative Electronic Courses in Universities
Vassily Yu. Kalachev, Guennady A. Ougolnitsky and Anatoly B. Usov
Southern Federal University, 8a, Milchakov St., Rostov-on-Don, 344090, Russia vkalachev@sfedu.ru; gaugolnickiy@sfedu.ru; busov@sfedu.ru
Abstract We built and investigated a two-level dynamic game theoretic model of control of the state promotion of innovative electronic courses in universities based on Cournot oligopoly. A federal state is the Principal, and universities competing a la Cournot are agents. The agents invest in the development of new electronic educative courses that is considered as their innovative investments. The Principal gives subsidies to the agents for the promotion of innovations. The agents play a dynamic game in normal form that results in a Nash equilibrium, and the Principal solves an inverse Stackelberg game (a Germeier game of the type r2t). We investigated different types of strategies: (1) uniform strategies for all agents; (2) type-dependent (agent efficiency-dependent) strategies; (3) action-dependent strategies. For a specific form of the model functions we found a solution in explicit form, and in the general case we used a method of qualitatively representative scenarios in simulation modeling. We analyzed the results by means of the individual and collective relative efficiency indices.
Keywords: Cournot oligopoly, inverse dynamic Stackelberg games, simulation modeling, university management.
1. Introduction
A problem of promotion of innovations is very actual and is discussed in literature. A concept of innovation funnel is considered in (Bonazzi and Zilber, 2014, Hakkarainen, 2014). A review of the mathematical models of economics with innovations is presented in (Makarov, 2009). In (Cellini and Lambertini 2002) they analyzed a dynamic oligopoly where firms invest to increase product differentiation. They compare the steady state solutions under the open-loop and the closed-loop Nash equilibrium. The authors' approach to the dynamic game theoretic modeling of the promotion of innovations in universities is proposed in (Malsagov et al., 2020, Kaluza et al., 2010). In this paper we emphasize a comparative analysis of the solutions that correspond to a selfish agents behavior, their hierarchical organization and cooperation (Ougolnitsky, 2022). For a quantitative evaluation of the different ways of organization we use individual and collective indices of the relative efficiency. As differs from (Kaluza et al., 2010), we pay the principal attention to different types of subsidies for the promotion of innovations in universities. A method of qualitatively representative scenarios in simulation modeling (Ougolnitsky and Usov, 2018) is used together with known numerical methods.
2. Problem Formulation
We consider a difference inverse Stackelberg game of the type Principal - agents. However, the equation of dynamics is a differential one. Agents are universities https://doi.org/10.21638/11701/spbu31.2023.05
competing a la Cournot: they develop electronic educative courses for sale. Resources allocated for this development are considered as innovative investments. The Principal is the federal state or its representative bodies (for example, Ministry of Education). The Principal exerts on the agents an economic influence (impulsion) by subsidies.
The model in the case of n agents has the following form:
- the Principal's payoff functional:
Jo = St I xxt I(xit)sit(xt)] + STyx ^ max (1)
t=i V ieN J
- Principal's budget constraints:
sit(xt) > 0; Y sjt(xt) < St; t = 1, 2,..., T; i e N = {1, 2,...,n} (2)
jeN
- agents' payoff functionals:
T 2
Ji = Y] St((D - aXt)xit--7-—-v - CiI(xit)+ (3)
2 (ri + d En=1;j=i I(Xjt)rj)
t
t=1
I(Xit)sit(xt)) ^ max
- agents' control constraints
0 < Xit < Xmax; i = 1, 2, ..., n; t = 1, 2, ...,T (4)
- an equation of dynamics
yt+1 = yt + kixit - myt; y(0) = yo (5)
ieN
Here J0, Ji - are payoffs of the Principal and the agents respectively; i e N; sit(xt)-a subsidy from the Principal to the i-th agent; St - an annual Principal's budget; xit - an output volume of the innovative product by the i-th agent in the moment of time t; xt = "=1 xit; xt = (x1t, x2t,..., xnt); ri - an agent's type that characterizes an efficiency of his technologies; D, x, xmax > 0; a, d > 0 - model parameters; S e (0,1) a discount factor; ci- constant agent's cost; I(xit) - indicator function;
I (xit) = 0, if xit = 0 and I (xit) = 1, if xit = 0;
yt — a general innovative level of the education system (a number of the used innovative products); m — a coefficient of decreasing of this level in the case when new innovative products are not developed; ki - an impact coefficient for the i-th product; y0- an initial value of the innovative level; T - a length of the game.
Thus, the model (1)-(5) is a difference inverse Stackelberg game (a Germeier game r2t) that is similar to a continuous version described in (Malsagov et al., 2020).
The Principal chooses her open-loop strategies with a feedback on control sit and reports them to the agents. Given the Principal's control mechanism, the agents choose their actions xit so that to attain a Nash equilibrium in their game in normal
form (3)-(4). The Nash equilibrium is treated as the agents' best response to the Principal's strategy. As the Principal anticipates the agents' best response, she chooses her strategies so that to solve the problem (1)-(2), (5) on the set of Nash equilibria in the game (3)-(4). If there are several Nash equilibria then the Principal uses the guaranteed result principle. The Principal's e -optimal strategy together with any best response of the agents form a solution in the inverse Stackelberg game (Germeier game r2t).
Notice that the agents in the model (1)-(5) are myopic, i.e. their payoff func-tionals may be rewritten as
T
Ji ^^ St Jit;
t=1 x2
Jit = (D - axt)xit--~y---^ - Cil(xit) + I(xit)sit(xt)
2 [n + f E;=i;j=i I(xit)rj)
and their optimal values do not depend on the state value, or on the solution of a differential equation (5). Therefore we can pass from an optimization problem for the functional (3) for the i-th agent to the optimization problem for T functions in the form
Jit ^ max; t = 1, 2, ...,T (6)
Each function (6) is maximized by the variable xt at a fixed moment of time t subject to the constraints (4). Thus, each agent solves T optimization problems (6),(4).
3. Nash Equilibrium
Consider a case of the indifferent Principal without her own objectives. Assume that the Principal's strategies are linear functions of the agent's actions: sit = sit(xit) = Yitxit; i = 1, 2,..., n . Then we receive a game of n agents (4)-(6) where a Nash equilibrium is built. For the i-th agent a maximal payoff is attained when xit = 0 , and then it is equal to zero, or when xit > 0 . Let us consider the latter case. Using a necessary first order condition
dJit
J =0; i = 1, 2,..., n, dxit
in the case of symmetrical agents
Ci = c; xit = xt; rit = rt; Yit = Yt; Ji = J; Jit = Jt; i = 1, 2,..., n we receive an equation for determination of their stationary controls d J
J = D - 2anxt - ( + f (Xt ) + Yt =0 (7)
dxt (r + f (n - 1)r)
Notice that
d 2Jt 1
1 = -2an- ----— < 0.
dx\ (r + P('n — 1)r)
Therefore, a solution of the equation (7) determines a maximum point, and if xt > 0 then an optimal control of the agent is given by the formula
0 = (D + Yt)r(1 + ft (n — 1)) Xt 1 + 2anr(1 + ft(n — 1))
In this case the agent's payoff is equal to
T = v (D + Yt)(r + rß(n - 1))2 f + _ ) + D + Yt \ J ° (1 + 2anr(1 + ß(n - 1)))2 V(D + Yt C) + 2r(1 + ß(n - 1)) J
t=i
T
t=i
Thus, equilibrium strategies and payoffs of the agents are determined by the formulas
x* =0, if x0 < 0 or At < 0 and x* = x0, otherwise (8)
T
J * = Y Jt; Jt = max (0, At) t=i
Therefore, the following proposition is proved.
Proposition 1. Formulas (8) determine a maximum point of the payoff functions (6) and payoffs of n symmetrical agents in a Nash equilibrium in the case of an indifferent Principal.
4. Cooperation of the Principal with Agents
In the case of cooperation of the Principal with n agents they form a grand coalition and solve together an optimal control problem with a payoff functional in the form
JC = ]T St Ixxt + Y \(D - axt)xit--t-—-^ - dl(*„)) J
t=i V ieN\ 2 {ri + ߣj=ij'=i1 (xit)ri) ))
+5TyT ^ max (9)
The maximum is searched by n functions (xit)"=1 subject to the constraints (4) and equation of dynamics (5). The game is reduced to an optimal control problem.
If controls of all agents are equal to zero: xit = 0; i = 1,2,..., n then the coali-tional payoff is equal to 5Ty0(1 — m)T.
Otherwise, for determination of the maximum in (9) a discrete Pontryagin maximum principle is used (Boltyanskii, 1978). An integrand in (9) is convex, the equation of dynamics is linear by the control variables that belong to a convex closed set. Therefore, for the solution of the problem (4),(5),(9) we can use a discrete Pontryagin maximum principle (Boltyanskii, 1978). A Hamilton function of the grand coalition has the form:
Ht (yt, At+i,xt) = St I xxt + (D - axt)xt - ^
ie^ 2 (Vi + ß E"=i;j=i rj) +At+i Y kiXit + (1 - m)yt ,
\i£N )
Ci
2
x
where At+i is a conjugate variable. From the necessary condition of extremum we receive the system of n equations i = 1,2,..., n
dHt _ t / , ^ ^ __ v^ / Xit
dx„ = ^ (x + D — 2ax— g {) I + A'+'k = 0 (10)
and for determination of the conjugate variable - a simple initial value problem
At = (1 - m)At+i; At = 5T,
therefore,
At = (1 - m)T-t5T.
In general case the system of equations (10) is solved numerically. For n=2 the system takes the form
dHt _ t f , ^ „ , , , xit
¿M x + D — 2a(xit + xat)--+ At+iki = 0;
dxit V ri + ftr2/
dHt _ xt {_ , n , N x2t
dx2t V r2 + ftri
Its solution gives
^ ( x + D — 2a(xit + x2t) — r ,fl„ ) + At+ik2 = 0. Ait — Bit
it 2a +1 + 1/(2a(ri + ftn ));
0 = A2t — B2t
x2t = 2a + 1 + 1/(2a(r2 + ftri))
where
Ait = X + D + k2(1 — m)T-t-i£T-t; A2t = X + D + ki(1 — m)T-t-i£T-t;
Bit = (2a + -L-) f X+D + ki" — -' ■
V r2 + ftri J \ 2a 2a
, , , , , , _ ko(1 - mT-'-UT-t B2t = 2a +-^-+
ri + ftr2 y \ 2a 2a
The found pair of points (x0t, x2t ) is a maximum point of the Hamilton function for positive controls xit,x2t. Really,
d2a * (.v. . i \ P „ „. d2a « i
«.ît = l2a + ÎTT^J = E < 0 axi; = —f l,2a + rrr^i ' = F <0;
d 2 H
t —2aJt = G < 0; 4 = EF — G2 > 0; E < 0.
dxitdx2t
Therefore, a maximum of the Hamilton function subject to the control constraints (4) is attained in one of the vector points
(x?t,x0t); (x0t, 0); (0,x°t); (0), 0); (11)
(x0t,xmax); (xmax,x2t; (xmax,xmax)
and the following proposition is proved.
Proposition 2. Formulas (11) determine a point of maximum of the Hamilton function in the case of cooperation of the Principal with two agents.
5. Solution of the Inverse Stackelberg Game (Germeier Game r2t)
From the point of view of the Principal her interaction with agents is described by an inverse Stackelberg game (Germeier game r2t). An algorithm of solution of this game is based on the approach proposed in (Ugolnitskii and Usov, 2014, 2016).
1. A strategy of punishment by the Principal of the agents who refuse to cooperate with her is calculated:
xit({4}i=i) = arg max Ji({sit}f=i, {xit}f=1);
0<Xit<Xmax
{sft}T=i = arg ™in „„ Ji({sit}T=i, {xit}T= i).
Sit>Q',Z_,ieN SitSSt
If an agent refuses to cooperate then his guaranteed payoff is equal to (i = 1, 2,..., n)
Li = Ji({sft }i=i, {xii }T=i) =
n,max min Ji({sit}T=i, {xit}T=i) 0<Xit <Xmax Sit>0; ¿J i£N Sit<St
and is determined by the formula similar to (8) when sit = 0; i = 1,2,..., n; t = 1,2,...,T.
2. An optimal control problem (1), (2), (4), (5) is solved with conditions
Li < Ji({sit}T=i, {xit}T=i); i =1,2,..., n. (12)
A maximum is searched by two grid functions {sit}"t=i, {xit}nt=i . Denote a solution of this optimal control problem by {sR}T=i, }T=i , where {sR}T=i is a strategy of reward of the i-th agent when he chooses }T=i.
3. The Principal reports to each agent a strategy with a feedback on his action:
sit = sR, if xit = xJt and sit = spt, otherwise
The condition (12) provides that for the agents a reward strategy is more profitable than a punishment strategy. Thus, the solution has the form ({sR}T=i, {xR}T=i).
A solution of the inverse Stackelberg game (Germeier game r2t ) is built numerically by means of the method of qualitatively representative scenarios in simulation modeling (QRS SM method) (Ougolnitsky and Usov, 2018).
The QRS SM method is based on the idea that for evaluation of the consequences of control impacts on a dynamic system it is sufficient to consider a small number of control scenarios that reflect qualitatively different variants of the impact. Assume that
Q = Si x ... x Sn x Xi x ... x Xn.
Here
n
Si = (si > 0; Y si < S); Xi = (xi > 0), i = 1, 2,...,n
i=i
are the sets of feasible controls of the Principal and agents. Definition (Ougolnitsky and Usov, 2018). A set
QRS = SQRS x XQRS = SQflS x S2QflS x... x SQRS x xQrs x XQRS x... x XQRS =
{(s, x) = (si,..., s„; xi,... ,x„); Sj G SQRS € S^; Xj G XQRS € XJ
is a QRS set in a Stackelberg game with precision A If:
(a) for any two elements (s,x)(j), (s,x)(j) G QRS | J0j) - J0j)| > A;
(b) for any element (s,x)(l) G QRS there is an element (s,x)(j) G QRS such that |j(l) - j(j)|< A.
An algorithm of solution of the inverse Stackelberg game (Germeier game r2t ) by means of the QRS SM method has the following form. 1. An initial set QRS(k) has the form (k = 0)
QRS(k) = (SQRS)(k) x (XQRS)(k);
(SQRS)(k) = (SQRS)(k) x (S2qrs)(k) x ...(SQRS)(k); (XQRS)(k) = (XQRS)(k) x (XQRS)(k) x ...(XQRS)(k);
(SQRS)(k) = {sik); s2k); s3k)};(xQRS)(k) = {x^; x2k); x«};
(k) = o- s(k) = s /9- s(k) = s • x(k) = 0- x(k) = x /9- x(k) = x 1 uj ^maxf ^max: uj ^2 ^maxj ^max?
where values smax,xmax are big enough and are chosen specifically for each control system.
2. The set QRS(k) contains 32N elements. All of them are checked for satisfaction of both conditions in the mentioned definition of a QRS set. If it is necessary then an initial set QRS(k) is reduced or extended by new elements.
3. A strategy of punishment of the agent who refuses to cooperate with the Principal is found. First, by enumeration of the strategies from the set (XQRS)(k) Nash equilibria for a given Principal's control NEQRS((SQRS)(k)) are found. Then a guaranteed payoff of the i-th agent who refuses to cooperate with the Principal is calculated:
L,P = max min Jj(sj,xj).
1 xi£NEQRS((SQRS)(k>) Sie(SQRS)(k>
4. By the complete enumeration of the qualitatively representative strategies of the Principal from (SQRS)(k) and the agents from (XQRS)(k) a maximum in the problem (1), (2), (4) with conditions Jj > L?(i = 1, 9, ...,n) is found.
The values that provide the maximum form a k-th approximation to the solution of the game. Denote them by (sR)(k), (xR)(k).
5. The QRS sets of the Principal and the agents (k := k + 1) are refined in the vicinity of the built equilibrium as follows.
if (st)(k-i) = s(1k-1), then sik) = s(1k-1); s^ = (s1k-1) + s2k-1))/9; s3k) = s2k-1).
If (st)(k-1) = s2k-1), the:1 s1k) = (s1k-1) + s1k-1))/9;s2k) = s2k-1); s3k) = (s2k-1) + s3k-1))/9.
If (st)(k-1) = s3k-1), then s1k) = s2k-1); s2k) = (s2k-1) + s3k-1))/2; s3k) = s3k-1).
New sets QRS(k) for the agents are built similarly.
If at an iteration we receive that (sR)(k) = (sR)(k-1); (xR)(k) = (xR)(k-1); i = 1, 2,..., n then a solution of the game by means of the QRS SM method is built. Otherwise, go to step 2 of the algorithm.
6. Numerical Calculations
We considered the following types of the Principal's strategies (her subsidies to the agents):
(a) uniform subsidies to all agents in a fixed moment of time. If an output volume of the innovative product for all agents is positive then Vi sit = s > 0 , otherwise Vi sit = 0;
(b) type-dependent (agent efficiency-dependent) strategies sit = si(rit); a linear dependency is used si(rit) = airit; constants ai are to be determined;
(c) action-dependent strategies sit = si(xit); a linear dependency is also used si(xit) = 0ixit; constants are to be determined.
All computer simulations were conducted on a personal computer with a processor AMD Ryzen 5 3550H with operative memory 8 Gb by means of an object-oriented programming language C++. An average time of one computer simulation for determination a QRS set is less than one second.
An analysis of the received results was based on the following indicators:
(1) a total discounted payoff of the Principal;
(2) values of the individual and collective relative efficiency indices (Ougolnitsky, 2022).
The collective relative efficiency indices demonstrate a need in a hierarchical control in a dynamic system. The closer are their values to one, the better the system is coordinated, and a hierarchical control by the Principal is less actual. In the computer simulations we varied the following parameter values:
1. x from 0.01 to 3;
2. D from 5 to 100;
3. A from 0.001 to 0.1 year/mln.rub.;
4. ri,2 from 0.5 to 50 thousand rub./year;
5. ci,2 from 50 to 1000 mln.rub./year;
6. P from 0.01 to 0.6;
7. m from 0.0001 to 0.11/year;
8. ki,2 from 0.001 to 0.051/year;
9. y0 from 30 to 500 mln.rub./year;
10. St from 100 to 500 mln.rub./year.
Input data for numerical calculations are presented in Table 1. The results of calculations for these data and T = 6; n = 2 for different control scenarios are given in Table 2. The upper index in the values J0k), j{fc), J2k) stands for a type of scenario, namely: (a) - uniform subsidies; (b) - type-dependent strategies; (c) -action-dependent strategies; j( C) denotes a payoff of the coalition of the Principal with agents in the case of their cooperation.
For a comparative analysis of the different scenarios of the Principal's control we used a system of the individual and collective relative efficiency indices (Ougolnitsky, 2022). The collective relative efficiency indices correlate the values of social welfare (a total payoff of all players) for different scenarios with the maximal value of social welfare that is attained in the case of cooperation of all players: SCI = En=0 Ji/JC. Here Ji is a payoff of the respective agent in a specific scenario (a),(b),(c) of the Principal's control, and JC is a cooperative payoff of the grand coalition in the case of cooperation.
The individual relative efficiency indices correlate payoffs of the agents in a specific scenario (a),(b),(c) of the Principal's control with their symmetrical coop-
Table 1. Input data for numerical calculations
N D ri r2 ci c2 yo ki k2 X a ß m
1 200 20 30 500 700 200 0.02 0.4 1 0.2 1 0.03
2 200 60 50 300 400 200 0.03 0.3 1 0.1 0.8 0.03
3 200 20 70 350 200 100 0.04 0.06 0.5 0.3 0.1 0.03
4 200 60 20 200 400 100 0.04 0.03 0.5 0.25 0.5 0.03
5 200 30 10 600 500 200 0.01 0.05 1 0.05 0.3 0.05
6 200 20 50 400 300 200 0.05 0.02 0.5 0.1 1.4 0.05
7 300 40 60 200 500 200 0.12 0.2 1 0.12 0.8 0.05
8 300 20 10 400 300 200 0.1 0.1 1.5 0.15 0.6 0.1
9 300 40 20 500 600 50 0.01 0.03 0.4 0.05 0.7 0.02
10 300 20 10 200 500 50 0.07 0.01 0.1 0.1 1 0.02
11 300 50 30 400 300 50 0.05 0.01 0.3 0.1 0.8 0.01
12 500 20 10 400 300 200 0.03 0.25 0.4 0.2 1 0.03
13 500 10 50 400 200 200 0.07 0.03 0.3 0.2 0.8 0.03
14 500 40 25 200 500 200 0.01 0.03 0.2 0.1 1.2 0.03
15 500 30 50 400 300 200 0.05 0.3 0.1 0.5 1.5 0.03
16 500 20 40 100 500 200 0.2 0.1 0.3 0.3 1 0.03
17 500 15 10 150 100 200 0.1 0.5 0.1 0.4 0.5 0.02
18 200 20 30 450 350 100 0.06 0.04 0.1 0.2 2 0.01
19 200 60 25 500 450 100 0.01 0.03 0.4 0.15 1 0.01
20 200 10 40 400 300 100 0.03 0.02 0.3 0.2 1 0.01
21 200 30 40 500 700 100 0.01 0.05 0.5 0.05 2 0.01
22 200 40 50 600 300 100 0.02 0.04 0.3 0.1 2 0.01
23 200 15 20 400 600 200 0.01 0.08 0.7 0.2 1 0.02
24 100 50 40 300 400 200 0.05 0.01 0.8 0.2 2 0.02
25 100 50 15 500 600 200 0.01 0.06 0.5 0.1 2 0.01
26 100 30 5 500 300 200 0.05 0.08 0.5 0.1 1 0.02
27 100 5 20 400 500 200 0.03 0.06 0.5 0.2 1 0.02
28 100 10 5 300 400 200 0.02 0.05 0.5 0.1 2 0.01
29 100 20 10 400 700 100 0.07 0.05 1 0.3 1 0.01
30 100 30 10 600 500 100 0.05 0.02 0.5 0.1 1.5 0.02
31 100 10 20 500 700 100 0.05 0.01 1.5 0.05 1.2 0.01
32 100 20 30 500 400 100 0.01 0.02 1 0.2 1 0.01
33 50 20 25 500 400 50 0.01 0.02 0.9 0.23 1.4 0.01
34 50 30 5 500 600 50 0.03 0.05 1.4 0.25 1.7 0.01
35 50 5 10 700 400 100 0.02 0.04 1.7 0.15 1.5 0.01
36 50 5 20 400 600 50 0.05 0.03 1.2 0.2 1.2 0.02
37 50 10 5 500 400 100 0.01 0.02 1.5 0.15 1 0.02
38 50 20 15 500 600 100 0.05 0.01 1.5 0.15 0.5 0.01
39 50 15 10 500 400 50 0.03 0.05 1 0.1 1 0.02
40 50 10 15 700 500 50 0.01 0.03 1.2 0.15 0.7 0.02
Table 2. Results of the numerical calculations
N Jc J (a) J0 J (a) J(a) J2 -O (J j (b) J1 -O J (c) J0 J (c) J1 J(c) J2
1 58756 1018 25434 24834 1260 25445 25065 1450 25475 25225
2 60027 868 27572 27371 1380 28042 27311 1300 27612 27662
3 53185 335 24320 24857 897 24331 25408 767 24361 25248
4 54733 285 25605 24984 767 26076 24995 717 25646 23375
5 62544 378 27345 27561 610 27576 27562 810 27386 27952
6 59429 238 27267 27540 640 27278 27961 670 27307 27961
7 89206 638 42566 41662 1140 45525 42139 1070 42557 42059
8 88442 568 41477 41748 880 41788 41748 1000 41518 42139
9 92185 938 42689 42381 560 43000 42391 670 42730 42771
10 89611 248 42784 41884 400 42935 41885 680 42825 42275
11 89601 408 42258 42255 620 42649 42576 840 42300 42946
12 146967 605 70684 70984 667 70735 71735 1036 70735 71735
13 146306 275 70734 71344 667 70721 71708 707 70702 71631
14 149337 145 72855 71957 472 73166 71972 577 72896 72348
15 137793 631 66273 66569 1043 66294 66960 1063 66314 66560
16 143530 608 70146 68946 930 70157 69257 1040 70188 69337
17 140595 1547 68371 68494 1660 68482 68496 1980 68412 6888
18 57034 548 25612 25906 790 25623 26136 980 25653 26296
19 56674 338 26214 26364 825 26685 26381 770 26255 26755
20 56674 358 25734 26034 670 25735 26345 790 25775 26425
21 62622 468 27724 27121 800 27746 27432 900 27765 27512
22 60146 408 26682 27580 830 26713 27971 840 26723 27971
23 57653 453 25701 25101 610 25707 25252 885 25742 25492
24 26546 408 11080 10782 830 11471 10813 840 11121 11173
25 30908 518 11962 11676 915 12353 11682 950 12003 12067
26 30312 493 11902 12502 720 12133 12498 925 11943 12893
27 27544 393 10659 10359 540 10655 10510 825 10700 10750
28 29975 518 12421 12159 585 12493 12155 950 12462 12550
29 25481 918 9184 8284 1070 9335 8285 1350 9225 8675
30 31258 643 11601 11859 875 11832 11860 1074 11643 12250
31 32984 678 12665 12051 830 11267 11202 1109 12704 12442
32 27306 468 10434 10734 710 10445 10965 900 10475 11125
33 8360 438 2491 2788 640 2501 2979 870 2532 3179
34 7944 838 2162 1886 1067 2163 2091 1270 2203 2283
35 10060 828 2972 3845 895 2968 3916 1260 3013 4236
36 9986 578 3180 2564 725 3176 2716 1009 3221 2956
37 10288 543 3509 3809 610 3580 3805 975 3550 4200
38 11842 768 3622 3309 760 3066 4011 1200 3663 3700
39 11512 518 4359 4659 630 4470 4660 950 4400 5050
40 9694 328 3025 3620 440 3026 3611 760 3066 4011
erative payoffs: K = (n + 1) Jj/JC; i = 0,1,2. It is supposed that all payoffs are non-negative. The received values of relative efficiency indices are presented in Table 3.
The last row of the Table 3 contains average values of the indices. Thus, we receive the following preference systems:
society: C >- (c) >- (b) >- (a);
Principal: C >- (c) >- (b) >- (a);
agents: (b) ~ (c) >- (a) >- C;
Thus, the whole society and the Principal prefer cooperation, and the agents (followers) prefer type-dependent or action-dependent subsidies.
Besides, the following conclusions are made.
1. A parameter x characterizes a dependency of the Principal's payoff on a total output volume of the innovative products. If its value increases then the Principal's payoff increases linearly for all types of subsidies. The agents' payoffs do not change.
2. If demand parameters D and a increase then the agents' payoffs increase exponentially. The Principal's payoff does not change.
3. If an agent's type changes (an efficiency of his technologies increases or decreases) then his payoff changes slightly. For example, if the efficiency increases twice then the payoff increases on 10% approximately. The Principal's payoff does not change.
4. If the agents' costs increase then their payoffs expectably fall.
5. Remind the parameters of the equation of state dynamics: m - a coefficient of decreasing of the innovative level; k - an impact coefficient for the i-th product. If these parameters change then the agents' payoffs do not change. However, the Principal's payoff decreases when m increases, and increases abruptly when k increase. Also, it increases together with an initial value of the innovative level.
7. Conclusion
We built and investigated a two-level control system aimed at promotion of innovations in the universities competing a la Cournot. The system is formalized as a difference inverse Stackelberg game (Germeier game r2i ) of the type Principal-agents with a differential equation of dynamics. Based on a discrete Pontryagin maximum principle, for a specific class of model functions in the case of an indifferent Principal we found analytically a Nash equilibrium in the game of agents in normal form. An algorithm of solution of the inverse Stackelberg game (Germeier game r2t) is proposed and implemented on the base of the method of qualitatively representative scenarios in simulation modeling. The received results allowed for some conclusions given above. The main conclusion is that the whole society and the Principal prefer cooperation, and the agents (followers) prefer type-dependent or action-dependent subsidies.
Universities are often myopic that we considered in the model. That's why a promotion of innovations advocates for the interested Principal who provides innovations by means of subsidies to the agents. The Principal's direct payoff may be quite small.
In the future we suppose to investigate the considered model in a cooperative dynamic game theoretic setup with different characteristic functions and to conduct a comparative analysis.
Table 3. The values of relative efficiency indices for different scenarios: (a) uniform strategies; (b) type-dependent strategies; (c) action-dependent strategies
N SCI(a) K0a), K(a), K(a) SCI(b) , , SCI(c) K0c), K(c), K2c)
1 0.87 0.05/1.3/1.27 0.88 0.06/1.29/1.28 0.88 0.07/1.3/1.29
2 0.93 0.04/1.38/1.87 0.95 0.07/1.39/1.36 0.94 0.06/1.38/1.39
3 0.99 0.02/1.31/1.4 0.95 0.05/1.37/1.43 0.95 0.04/1.37/1.42
4 0.93 0.02/1.4/1.4 0.95 0.04/1.43/1.37 0.91 0.04/1.41/1.28
5 0.88 0.02/1.31/1.32 0.89 0.03/1.32/1.32 0.9 0.04/1.31/1.34
6 0.93 0.01/1.38/1.39 0.94 0.03/1.38/1.41 0.94 0.03/1.38/1.41
7 0.95 0.02/1.43/1.40 0.99 0.04/1.53/1.42 0.96 0.04/1.43/1.41
8 0.95 0.02/1.41/1.42 0.95 0.03/1.42/1.41 0.96 0.04/1.43/1.41
9 0.97 0.03/1.39/1.38 0.93 0.02/1.4/1.38 0.93 0.02/1.39/1.39
10 0.95 0.01/1.43/1.4 0.95 0.01/1.44/1.4 0.96 0.02/1.43/1.42
11 0.95 0.01/1.42/1.41 0.96 0.02/1.43/1.43 0.96 0.03/1.42/1.44
12 0.97 0.01/1.44/1.45 0.97 0.01/1.44/1.46 0.98 0.02/1.44/1.46
13 0.97 0.01/1.45/1.44 0.97 0.01/1.44/1.46 0.97 0.01/1.44/1.46
14 0.97 0.01/1.46/1.45 0.92 0.01/1.47/1.45 0.98 0.01/1.46/1.45
15 0.97 0.01/1.44/1.45 0.97 0.02/1.48/1.46 0.97 0.02/1.44/1.45
16 0.97 0.01/1.47/1.44 0.98 0.02/1.47/1.45 0.98 0.02/1.47/1.45
17 0.98 0.03/1.46/1.47 0.98 0.04/1.46/1.47 0.99 0.04/1.46/1.48
18 0.91 0.03/1.35/1.36 0.92 0.04/1.35/1.37 0.93 0.02/1.35/1.38
19 0.93 0.02/1.38/1.39 0.95 0.04/1.41/1.4 0.95 0.04/1.39/1.42
20 0.92 0.02/1.36/1.38 0.93 0.04/1.36/1.39 0.95 0.04/1.39/1.42
21 0.88 0.02/1.33/1.31 0.89 0.04/1.33/1.38 0.9 0.04/1.33/1.32
22 0.91 0.02/1.33/1.35 0.92 0.04/1.33/1.4 0.92 0.04/1.33/1.4
23 0.89 0.02/1.34/1.31 0.89 0.03/1.34/1.31 0.89 0.05/1.34/1.32
24 0.84 0.05/1.25/1.22 0.87 0.09/1.3/1.22 0.87 0.09/1.26/1.26
25 0.78 0.05/1.16/1.13 0.82 0.09/1.2/1.13 0.81 0.09/1.17/1.17
26 0.82 0.05/1.18/1.21 0.83 0.07/1.2/1.24 0.85 0.09/1.18/1.28
27 0.78 0.04/1.16/1.13 0.79 0.06/1.16/1.14 0.81 0.09/1.16/1.17
28 0.84 0.05/1.24/1.21 0.84 0.06/1.25/1.22 0.87 1.1/1.25/1.26
29 0.72 0.11/1.08/0.99 0.74 0.13/1.1/0.98 0.76 0.16/1.09/1.02
30 0.77 0.06/1.11/1.09 0.79 0.08/1.14/1.14 0.8 0.1/1.1/1.18
31 0.77 0.06/1.15/1.11 0.71 0.08/1.02/1.02 0.8 0.1/1.16/1.13
32 0.79 0.05/1.15/1.19 0.81 0.08/1.15/1.2 0.82 0.1/1.15/1.22
33 0.68 0.16/0.89/1 0.73 0.23/0.9/1.07 0.79 0.31/0.92/1.14
34 0.63 0.32/0.82/0.71 0.67 0.4/0.82/0.79 0.72 0.48/0.83/0.86
35 0.75 0.25/0.89/1.15 0.77 0.27/0.89/1.17 0.85 0.38/0.9/1.2
36 0.63 0.17/0.96/0.77 0.66 0.22/0.95/0.82 0.72 0.3/0.97/0.9
37 0.76 0.16/1.02/1.11 0.74 0.18/1.04/1.11 0.85 0.2/1.04/1.22
38 0.65 0.19/0.92/0.84 0.66 0.19/0.78/1.02 0.72 0.3/0.93/0.94
39 0.83 0.13/1.14/1.21 0.85 0.16/1.16/1.21 0.9 0.25/1.15/1.32
40 0.72 0.1/0.94/1.12 0.73 0.14/0.94/1.12 0.81 0.24/0.95/1.24
Average 0.86 0.06/1.25/1.24 0.87 0.07/1.27/1.26 0.89 0.1/1.26/1.3
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