CC BY
3David W. K. Yeung*
Center of Game Theory, Hong Kong Baptist University and
Center of Game Theory, St Petersburg State University
Abstract Economic analysis no longer treats the economic system as given since the appearance of Leonid Hurwicz’s pioneering work on mechanism design. The failure of the market to provide an effective mechanism for optimal resource use will arise if there exist imperfect market structure, externalities, imperfect information or public goods. These phenomena are prevalent in the current global economy. As a result inefficient outcomes continue to emerge under the conventional market system. Cooperative game theory suggests the possibility of socially optimal and group efficient solutions to decision problems involving strategic actions. This lecture focuses on cooperative game-theoretic design of mechanisms for optimal resource use.
Crucial features that are essential for a successful mechanism - individual rationality, group optimality, dynamic consistency, distribution procedures, budget balance, financing, incentives to cooperate and practicable institutional arrangements - are considered. Finally, cooperative game-theoretic mechanism design is used to establish the foundation for an effective policy menu to tackle sub-optimal resource use problems which the conventional market mechanism fails to resolve.
Keywords: Cooperative differential games, mechanism design, optimal resource use
Emile de Laveleye (1882): ” Political economy may... be defined as the science which determines what laws men ought to adopt in order that they may, with the least possible exertion, procure the greatest abundance of things useful for the satisfaction of their wants, may distribute them justly and consume them rationally.”
1. Introduction
Economic analysis no longer treats the economic system as given since the appearance of Leonid Hurwicz’s (1973) pioneering work on mechanism design. The term “design” stresses that the structure of the economic system is to be regarded as an unknown. New mechanisms are like synthetic chemicals: even if not usable for practical purposes, they can be studied in a pure form and so contribute to the understanding of the difficulties and potentialities of design. The design point of view enlarges our vision and helps economics avoid a narrow focus on existing institutions. The failure of the market to provide an effective mechanism for optimal
* The author is grateful to the research support by HK Research Grants Council CERG-HKBU202807, European Commission TOCSIN Project RTD REG/I.5(2006)D/553242, and HKBU Strategic Development Fund 03-17-224.
resource use will arise if there exist imperfect market structure, externalities, imperfect information or public goods. These phenomena are prevalent in the current global economy. As a result not only inefficient outcomes like over-extraction of natural resources had appeared but gravely detrimental events like catastrophe-bound industrial pollution had also emerged under the conventional market system.
As Hurwicz (1973) pointed out that a major impetus was given to the design of mechanisms by the developments of:
i) activity analysis and linear programming (including the simplex method)— Dantzig, Kantorovitch, Koopmans;
ii) game theory, including the iterative solution procedures—von Neumann and Morgenstern, George Brown, Julia Robinson;
iii) discoveries concerning the relationships connecting programming (linear or nonlinear), two-person zero sum games, and the long known Lagrange multipliers—Gale, Kuhn, Tucker.
Moreover, he stated: “While in economics one deals with goal conflicts due to multiplicity of consumers, linear and nonlinear programming models usually presuppose a single well-defined objective function to be, say, maximized, i.e., a situation corresponding to an economy with a single consumer. So it is not surprising that the mechanisms designed under the influence of programming theory dealt to a large extent with one-objective-function problems and thus failed to face the crucial issue of goal conflict.”
Hurwicz continued to say: “it is evident that the incentive structure is largely determined by what the participants can achieve for themselves by their free actions; this in turn depends on such institutional phenomena as private property, rules for the distribution of profits, or the freedom not to trade. A tool appropriate for the analysis of such phenomena is the characteristic function of a game defined by von Neumann and Morgenstern. Shapley and Shubik carried out a study of different institutional property arrangements, including feudalism, sharecropping, and the village commune, by constructing the corresponding characteristic functions and exploring the different versions of game solutions (von Neumann-Morgenstern solutions, the core, the Shapley value). Thus a significant step is taken toward a formalization of the distributional aspects of the economic system.”
Cooperative games suggest the possibility of socially optimal and group efficient solutions to decision problems involving strategic actions. This lecture focuses on cooperative game-theoretic design of mechanisms for optimal resource use. Since resource use is often a dynamic process we concentrate on the design of mechanisms involving an intertemporal framework. Crucial features that are essential for a successful mechanism - individual rationality, group optimality, dynamic consistency, distribution procedures, budget balance, financing, incentives to cooperate and practicable institutional arrangements - are considered. Finally, cooperative game-theoretic mechanism design is used to establish the foundation for an effective policy menu to tackle sub-optimal resource use problems which the conventional market mechanism fails to resolve.
The lecture is organized as follows. To formulate dynamic cooperative game-theoretic mechanism design, we first present the basic setting of cooperative differential Games, and the notions of group optimality and individual rationality. This is done in Section 2. The concepts of dynamic stability which is essential to the sustainability of a mechanism design are discussed in Section 3. The correspond-
ing payoff distribution procedures leading to the realization of dynamic stability solutions are derived in Section 4. Section 5 considers noncooperative equivalence imputation as a benchmark of allocation in optimal resource use mechanism design. Mechanism design for global environmental management is discussed in Section 6. The impact of financial constraint and irrational behavior on mechanism design is explored in Section 7. Concluding remarks are given in Section 8.
2. Cooperative Differential Games, Group Optimality and Individual Rationality
To formulate dynamic cooperative game-theoretic mechanism design, we first have to present the basic setting of cooperative differential Games, and the notions of group optimality and individual rationality.
2.1. Basic Settings of Cooperative Differential Games
Differential games study a class of decision problems, under which the evolution of the state is described by a differential equation and the players act throughout a time interval. In particular, in the general n-person differential game, player i seeks to maximize its objective:
max { / g1 [s,x(s),ui(s),u (s),''' , un(s)]ds + qi (x(T)) }, (2.1)
u J to
for i € N = {1, 2,...,n}, subject to the deterministic dynamics
x(s) = f [s,x(s),u1 (s),u2 (s), ••• ,un (s)], x(to )= xo, (2.2)
where x(s) € X C Rm denotes the state variables of game, and ui € Ui is the control of player i, for i € N.
In the case when the terminal horizon T approaches infinity an autonomous game structure with constant discounting will replace (2.1) and (2.2). In particular, the game becomes:
/•^
gi[x(s),u1 (s),u2 (s), ••• ,un(s)] x exp[—r(s — t0 )]ds, for i € N, (2.3)
to
subject to the deterministic dynamics
x(s) = f [x(s),ui (s),u2 (s), ••• ,u„(s)], x(to )= xo, (2.4)
where r is a constant discount rate.
Basar and Olsder (1995) provided a comprehensive overview of the analysis of zero-sum and non zero-sum noncooperative differential games.
Consider the case when the players agree to act cooperatively and play a cooperative game. The agreements on how to act cooperatively and allocate the cooperative payoffs constitute the solution optimality principle of a cooperative scheme. In particular, the solution optimality principle for a cooperative differential game includes:
(i) an agreement on a set of cooperative strategies/controls {u^(s),u2(s), ■ ■ ■ ,ujj(s)} for s € [t0,T], which would also determine the players’ payoffs in the case when payoffs are nontransferable; and
(ii) a mechanism to distribute total payoff among players with the players’ cooperative payoffs being {£ 1(s),£2(s), ••• ,£n(s)} for s € [t0,T ], in the case when payoffs are transferable.
2.2. Group Optimality and Individual Rationality
An essential element of a cooperative solution is group optimality, which ensures that all potential gains from cooperation are captured. Failure to guarantee group optimality leads to condition where there would be incentive to deviate from the agreed upon solution plan in order to extract the unexploited gains. Consider first the cooperative differential game with transferable payoffs (2.1) and dynamics (2.2) which solutions are based on group optimality. To secure group optimality the players seek to maximize their joint payoff by solving the following optimal control problem:
{t T n'
/ / gj [s,x(s),ui(s),u2(s)
j to j=
subject to (2.2).
Let { [^ (t, x), ^2 (t, x), ■ ■ ■ , (t, x)], for t € [to, T] } denote a set of controls
that provides an optimal solution (assuming its existence) to the control problem
(2.5) - (2.2).
Substituting this set of control into (2.2) yields the dynamics of the optimal (cooperative) trajectory as;
x(s) = f [s,x(s),^2(s,x(s)),^2(s,x(s)), ■ ■ ■ ,^n(s,x(s))], x(to) = xo. (2.6)
Let x2 (t) denote the solution to (2.6). The optimal trajectory {x2 (t)}J=to can be expressed as:
x2(t) = xo + [ f [s,x2(s),^2(s,x2(s)),^j(s,x2(s)), ••• ,^2 (s,x2 (s))]ds. (2.7)
to
For notational convenience, we use the terms x2 (t) and xt2 interchangeably in case where there is no ambiguity. The optimal level of joint payoff along the cooperative trajectory can be expressed as:
T n
W(t,x2) = /t Z gj[s,x2(s),^i(s,x2(s)),^2(s,x*(s)) j=1
n
+ £ qj(x2(T)). j=1
Note that group optimality will be guaranteed only if the agreed upon optimal strategies [^1*(t,x),^2 (t,x), ••• , ^n (t,x)] are adopted throughout the game horizon [to,T]. Dockner and J0rgensen (1984), Dockner and Long (1993), Tahvonen (1994), Maler and de Zeeuw (1998) and Rubio and Casino (2002) presented cooperative solutions satisfying group optimality in differential games.
Individual rationality is another essential element of a cooperative game solution. If the players in a game wish to make an agreement to share the benefits of cooperation, the axiom of individual rationality states that no player is willing to accept an agreement that will give him less payoff than what he could obtain
, ••• (s,x*(s))]
(2.8)
, ••• ,u„(s)] ds + ¿ qj (x(T )), (2.5)
by rejecting to participate in the cooperative solution. In games evolving over time, resolving the problem of individual rationality may not be easy. The reason is that individual rationality may fail to apply when the game has reached a certain position, despite the fact that it was satisfied at the outset. Maintaining individual rationality and group optimality throughout the game horizon are two essential factors for dynamic stability in cooperation to arise.
Consider the transferable payoff cooperative game (2.5)-(2.6). Along the optimal trajectory |x*(i)}T=t0 in (2.7), player i would receive over the time interval an amount equaling
J gl[s,x*(s),^i(s,x*(s)),^2(s>x*(s))>••• ^n(s,x*(s))]ds + ql(x*(T)). (2.9)
To secure individual rationality an instantaneous transfer or side payment n(s) at time s is given to player i so that the actual cooperative payoff offered to player i over the duration [t,T] becomes
f(t,x2) = ItT {gl [s,x*(s),^i*(s,x*(s)),^2 (s,x*(s)), ••• ,c(s,x*(s))] (210)
+n(s)}ds + ql(x2(T)). ^ ' >
Individual rationality requires the cooperative payoff for each player to be no less than his noncooperative payoff throughout the game horizon, that is
£l(t,x2) > Vl(t,x2), for i G N and t £ [to,T], (2.11)
where Vl(t,x2) is player i’s payoff value function in the noncooperative differential
game:
max | J gl [s,x(s),M1(s),u2 (s), ••• ,un(s)] ds + ql (x(T)) |, for i G N, subject to
x(s) = f [s,x(s),Mi(s),«2 (s), ••• ,u„(s)], x(t) = x2. (2.12)
For group optimality to be realized, it is required that
n
(t,x2) = w (t,x2), (2.13)
j=1
where W(t,x2) is the value function of the optimal control problem:
P T n n I
J [s,x(s),ui(s),u2(s),••• ,un(s)]ds +53qj(x(T)), (2.14)
max
Ul,U2,-” ,Un
j = l j=l
subject to (2.10).
An optimality principle under which the players agree to act to maximize joint profit (2.5) and allocate cooperative payoff {£ 1(t,x2),£2(t,x2), ••• , £n(t,x2)} fulfilling (2.11) and (2.13) yields a cooperative solution which satisfies group optimality and individual rationality.
The majority of cooperative differential games adopt solutions satisfying the essential criteria for dynamic stability - group optimality and individual rationality. Haurie and Zaccour (1986 and 1991), Kaitala and Pohjola (1988, 1990 and 1995), Kaitala et al (1995) and J0rgensen and Zaccour (2001) presented classes of transferable-payoff cooperative differential games with solutions which are required to satisfy group optimality and individual rationality are satisfied.
One way to incorporate stochastic elements in differential games is to introduce stochastic dynamics. A stochastic formulation for differential games of prescribed duration involves a vector-valued stochastic differential equation
dx(s) = f [s,x(s),M1 (s), (s), ■ ■ ■ , un(s)] ds + ct[s, x(s)]dz(s). x(to) = xo. (2.15)
which describes the evolution of the state, and n objective functions
u2(s),...,un(s)] x exp[-/t|! r(y)dy],ds , ,
t (2.16)
+ exp[- fto r(y)dy] ql(x(T))J , for i G N,
with Et0 {■ } denoting the expectation operation taken at time t0, a[s,x(s)] is a mx mi matrix and z(s) is a mi-dimensional Wiener process and the initial state x0 is given. Let Ü[s, x(s)] = ct[s, x(s)] a[s, x(s)]’ denote the covariance matrix with its element in row h and column Z denoted by ÜhZ[s,x(s)]. Moreover, E[dzw] = 0 and E [dzw dt] = 0 and E[(dzw )2 ] = dt, for w G [1,2, ••• ,m1]; and E[dzw dzk ] = 0, for w G [1, 2, ■ ■ ■ , m1 ], k G [1, 2, ■ ■ ■ , m1 ] and w = k.
2.3. Games with Non-transferable Payoffs
In order to maintain Pareto (1909) optimality when payoffs are not transferable the players seek to maximize the payoff (see Leitmann (1974)):
EtA fl0 g [s,x(s),ui (s)
I T n
max < f. ^2 aj [s,x(s),ui (s),U2 (s), ■■■ ,un(s)] ds +
ui,u2,••• I t0 j=i j
n
E aj qj (x(T))
j=i
subject to (2.2), where a = {a1, a2, ■ ■ ■ , an} is a vector of weights.
A necessary condition for individual rationality to hold is that at the initial time to, £(a)i (to ,x0) must be greater than Vl(t0,x0) for all players i G N for the chosen
weights a. Let { [^1a(t,x),^2a (t,x), ••• ,^n(t,x)], for t G [t0 ,T ] } denote a set of
controls that provides an optimal solution to the control problem (2.17) - (2.2), if the vector of weights a is chosen. Once again group optimality will be guaranteed only if the agreed upon optimal strategies [^1“(t,x),^2a(t,x), ■ ■ ■ ,^n(t,x)] are adopted throughout the game horizon [t0, T]. The optimal trajectory {xa(t)}T=to can then be expressed as:
xa (t) = x0 + i f [s,xa (s),^“ (s,xa (s)),^2“(s,xa (s)), ••• ,^n* (s,xa (s))] ds.
.o
(2.18)
For notational convenience, we use the terms xa(t) and xf interchangeably.
Along the optimal trajectory {xa(t)}T=to, the payoff of player i receive over the time interval [t,T], for t G [t0,T], becomes
£(a)i(t,x°) = J gi[s,x°(s),^°(s,x“(s)),^2(s,x“(s)),••• ,^a(s,x“(s))]ds
+q* (x2 (T)). (2.19)
While at the initial time t0, £(a)i (t0, x0) is greater than V* (t0, x0) for all players there is no guarantee that £(a)i(t,x0)> V* (t,x0), for all i G N and t G [t0 ,T] along the optimal trajectory {x2 (t)}T=to. Most existing cooperative differential games with nontransferable payoffs offer solutions which satisfy group optimality throughout the game horizon but not individual rationality. Threats and monitoring schemes are used to deter players deviating from the cooperative strategies as the game proceeds. Leitmann (1974 and 1975), Tolwinski et al (1986), Hamalainen et. al (1986), Haurie and Pohjola (1987), Gao et al (1989), Haurie (1991) and Haurie et al (1994) presented solutions satisfying group optimality and individual rationality at the initial time to cooperative differential games with nontransferable payoffs. In addition, threats are sometimes used to ensure that no players will deviate from the agreed-upon cooperative strategies throughout the game horizon (see Hamalainen et al (1986) and Tolwinski et al (1986)).
In particular, an optimality principle under which the players agree to act to choose a vector a throughout the game horizon to maximize (2.17) and the chosen vector of weights a leads to the satisfaction of
£(a)i(t,x°) > V*(t,x°), fori G N and t G [t0,T], (2.20)
yields a cooperative solution which satisfies group optimality and individual rationality throughout the game horizon.
In order to verify that individual rationality in a cooperative scheme holds along the optimal trajectory, we have to derive individual player’s payoff functions under cooperation
£(a)i(t,xf) for i G N and t G [t0,T]. One way is evaluate the integral in (2.19). Another way is to obtain an analytic solution of £(a)i(t,x0). Yeung (2004) showed the derivation of such analytic solutions by noting that for At ^ 0, we can write:
/t+At
gl ^(s,x“ (s)),^° (s,x“ (s)) ■ ,^a(s,x“(s))] ds
+? * (t + At, x° + Ax°), (2.21)
where 4x0*=f [s, x2(s), ^ a(s, xa(s)), ^2 (s, xa(s)), ■ ■ ■ , '^0, (s, xa(s))] A t.
Applying Taylor’s Theorem, we have
£(2)i (t,x2) = g* [t,x2 ,V> a (t,x2 ),^2a (t,x2), ••• ,c (t,xa )]At + ? *(t,x0)
a )f [s,xa (s),^ a (s,xa (s)),^a (s,x“ (s)), ■
C(s,xa(s))]
Canceling terms, performing the expectation operator, dividing throughout by At and taking At ^ 0, we obtain:
-£t(a)i(t,x2) = gi[t,xa^ a(t,x2 ),^° (t,x2% ••• ,^a(t,x°)] + ^ (t,x°)
(2.22)
xf [s,x2(s),^ f(s,xa(s)),^a (s,x2(s)), ■ ,^° (s,x2(s))] .
Boundary conditions require:
£(2)i (T,x0 )= q*(x2). (2.23)
Therefore if there exist continuously differentiable functions
£(a)i(t,x0) : [t0,T] x Rn ^ R satisfying (2.22) and (2.23)
then £(a)i (t,x2) gives player i’s cooperative payoff over the interval [t,T] with a being the cooperative weight .
Given explicit functions of £(a)i (t, x0) for i G N, one can verify individual rationality readily by checking the condition £(a)i(t,x0)> V1 (t,x0), for i G N and t G [t0,T]. Yeung and Petrosyan (2006a) presented solutions satisfying group optimality and individual rationality throughout the entire game horizon to cooperative differential games.
Finally, details on necessary conditions for Pareto optimal controls in cooperative differential games can be found in the work developed by Vincent and Leitmenn (1970), Leitmann et al (1972), Stalford (1972), Blaquiere et al (1972) and Leitmann and Schmitendorf (1974). Particulars concerning sufficient conditions are depicted in Leitmann and Schmitendorf (1963), Leitmann and Stalford (1971) and Leitmann (1974).
3. Dynamic Stability and Mechanism Design
Though group optimality and individual rationality constitute two essential conditions for dynamically stable cooperation, a stringent condition - time consistency -is required to achieve dynamic stability. Under a time-consistent solution, the solution optimality principle determined at the outset must be maintained and remain optimal throughout the game along the optimal state trajectory. In other words, time consistency of solutions to any cooperative differential game involved the property that, as the game proceeds along an optimal trajectory, players are guided by the same optimality principle at each instant of time, and hence do not possess incentives to deviate from the previously adopted optimal behavior throughout the game.
The question of dynamic stability in differential games has been rigorously explored in the past three decades. Haurie (1976) raised the problem of instability when the Nash bargaining solution is extended to differential games. Petrosyan (1977) formalized the notion of dynamic stability in solutions of differential games. Petrosyan and Danilov (1979 and 1982) introduced the notion of “imputation distribution procedure” for cooperative solution. Petrosyan (1991 and 1993b)) studied the time consistency of optimality principles in non-zero sum cooperative differential games. Petrosyan (1993a) and Petrosyan and Zenkevich (1996) presented a detailed analysis of dynamic stability in cooperative differential games, in which the
method of regularization was introduced to construct time consistent solutions. Yeung and Petrosyan (2001) designed time consistent solutions in differential games and characterized the conditions that the allocation-distribution procedure must satisfy. Petrosyan (1995b and 2003) employed the regularization method to construct time consistent bargaining procedures.
Dynamic stability is essential to the sustainability of a mechanism design. In this section, we review the notion of time consistency in cooperative differential games with transferable payoffs and survey transferable payoffs games with time consistent solutions.
3.1. Time-consistent Solutions in Games with Transferable Payoffs
For the sake of notational uniformity in the following exposition we identify discounting explicitly and express the objective of player i in (2.1) as:
m„ax{/: gi[s,x(s),u1 (s),u2 (s), ••• ,u„(s)] x exp[—r(y)dy] ds + exp[ /to r(y)dy] q*(x(T))} for i G N,
where r(s) is the discount rate at time s. In the case where there is no discounting r(y) = 0 for y G [t0,T].
At time t0 with the state being x0, the players are facing the cooperative
game (3.1)-(2.2), which is denoted by r(x0 ,T — t0). In the start of the game, that
is at time t0, the players agree to adopt a solution optimality principle 0(x0, T —10) under which the players would maximize joint cooperative payoff and allocate to the player i an agreed-upon cooperative payoffs over the period [t0, T].
After the cooperative game has been played for some time and at time instant t G (t0 , T], the players are facing the game
max j/t° g*[s,x(s),ui (s),u2 (s), ••• ,u„(s)] x exp[— /ts r(y)dy] ds Ui (3.2)
+ exp[— JtT r(y)dy] q%(x(T)) j for i G N,
subject to
x(s) = f [s,x(s),ui(s),u2 (s), ••• ,u„(s)], x(t) = x*. (3.3)
We use r(x*,T — t) to denote the cooperative game (3.2) - (3.3).
Adopting the agreed upon solution principle at time t yields 0(x*, T — t) for the game r (x*, T — t).
At time t0 with the state being x*, according to optimality principle
0(x0, T—10) the payoff over the period [t0, T] imputed to the player i is £(to)l(t0, x0), for i G N .At time t with the state being x*, if the players apply the originally agreed upon solution optimality principle 0(x*, T — t) in the game r (x*, T — t) the payoff over the period [t,T] imputed to the player i is:
£(t)l(t,x*) for i G N and t G [t0,T]. (3.4)
The vectors £(t)(t,x*) = [£(t)1(t,x*), £(t)2(t,x*),... ,£(t)n(t,x*)], for t G [t0,T] is a
valid imputation vector which satisfies the following necessary condition.
Condition 3.1
(i) £ £(t)j (t,x *) = W(t,x *), j=1
(ii) £(t)*(t, x*)> V(t)*(t,x*), for i G N and t G [t0,T]. □
Note that (i) ensures Pareto optimality while (ii) guarantees individual rationality.
A payoff distribution over time must be formulated so that the solution imputations in (3.4) can be realized. Let the vectors BT (s) = [BJ (s),BJ (s), ••• ,BJJ (s)] denote the “instantaneous” rate of payments to the players at time instant s G [t, T] in the cooperative game r (x*, T—t). A terminal value of ql (x^) is received by player i at time T.
In particular, BJ(s) and ql(xT) constitute a payoff distribution for the game r(x*,T — t) in the sense that £(t)i(t, x*) equals:
/T BT (s) exp [- /Ts r(y)dy] ds+
ql(xf) exp -/tT r(y)dy
c(r) = .
(3.5)
for i G N and t G [t0,T].
Moreover, for i G N and t G [t, T], we use the term £(t)l(t,x *) which equals
if BT (s)exP [-/ tS r(y)dy ds+
q*(xf) exp -/tT r(y)dy
x(t)
(3.6)
to denote the present value of player i’ cooperative payoff over the time interval [t,T], given that the state is x* at time t G [t,T], for the game r(x*,T — t).
From (3.5) and (3.6) one can readily observe that the solution optimality principle 0(x * ,T — t) would remain in effect only it assigns an imputation vector £(t)(t,x*) = [£(t)1 (t,x*),£(t)2(t,x*), ••• ,£(t)n(T,x*)] to the subgame r(x*,T — t) satisfies the condition that
£(io)i(r, x* ) exp
'■(y)dy
to
= e(T }Í(7
(3.7)
for i G N and t G [t0,T].
Crucial to the analysis is the formulation of a payment distribution mechanism that would lead to a time consistent solution. Section 4 will give an account on the derivation of such payment distribution mechanisms.
3.2. Time-consistent Solutions with Nontransferable Payoffs
In the case when payoffs are nontransferable, achieving time consistency is a much more difficult task. Haurie (1976) pointed out the problem of instability when the Nash bargaining solution is extended to differential games. Consider the nontransferable payoff version of the game 3.1-2.2, which we denote by -T(x0,T — t0).
In the game T^(x0 , T — t0), let a0 be a vector of weights selected according to an agreed upon optimality principle 0(x0 , T — t0). The optimal trajectory
x
T
{ i“° (t)} can be expressed as:
L ) t=tn
) / N
(t) = xo +
+ / tn / [s,xT" (s),^iin} a (s,xT“ (s)),^^a (s,x“u (s)), ••• (s,x^ (s))] ds.
n (3-8)
Along the optimal trajectory {xT (i)}t=tn, the payoff of player i receive over the time interval [t,T], for t G [to, T], becomes
5,°"’(t, x?°)
= it (s),"/'(t°)a (s,xa"(s)),^2^* (s.x^(s)), ■ ■ ■ (s,xa"(s))]
(to )a0
x exp
P s i T
'IT 1 ds + exp - r(y)dy
t0 t0
q'(xa (T)), for * G N. (3.9)
The chosen vector of weights a0 must satisfy individual rationality so that: (tn)(t,xa ) > V(tn)i(t,xT ), for i G N and t G [t0,T].
After cooperating for some time, at time instant t, the state becomes xT . In the
Qn
game ,T(xT ,T — t), let aT be the vector of selected weights according the original agreed upon optimality principle 0(xT , T — t).
The payoff of player i receive over the time interval [t,T], for t G [t,T], becomes
_TT(t) / . T
c(T )(t,xa ) =
ZtT g'|s,x^ (s),<)a (s,x^ (s)),^(T)a (s,x^ (s)), ••• ,^nr)a (s,x^ (s))]
',(r )
!,(r )c
x exp where
C s i T
)d 'IT - ds + exp - r(y)dy
j T </ T
ç'(x“T (T)), for * G N, (3.10)
(t) = x? +
+/T f |s,x^ (s),^(r)a (s,x“T (s)),^“ (s,x“T (s)), ••• ,^nJ“ (s,x“T (s))] ds.
(3.11)
Similar to the case when payoffs are transferable, if a time consistent solu-
/,(r ) <
tion assigns an imputation vector ?T (t)(t, xT ) = [?T (t)(t, xT ), ?T (T)(t, xT )
, ••• ,?T (t )(t, xT )] to the subgame r (xT ,T — t ), for t G [t0, T ], the following condition must be satisfied:
Tn(in)/ a
xT
for i G N and t G [t0, T].
V,xT )exp /tT0r(y)dy = çf (r)(r,x? ),
(3.12)
a
x
u
a
x
(t ) t0
For group optimality to be achievable, the cooperative controls [^ (n) (s,
xT° (s)),^(in) T (s,xT° (s)), ••• ,^iin) T (s,xT° (s))] must be adopted throughout time interval [t0,T]. Hence a subgame consistent solution optimality principle 0(x0,T — t0) which chooses a0 in the game P(x0,T — t0) will choose a0 again
in any subgame P(xT , T — t) the weight a0 chosen according to the solution optimality 0(xT ,T — t). Yeung and Petrosyan (2006a) showed that a time consistent solution to the nontransferable payoffs game P(x0, T —10) requires then satisfaction of the following theorem.
Theorem 3.1. A solution optimality principle under which the players agree to choose the same weight aT = a0 in all the subgames P(xT ,T — t) for t G [t0,T] and
0 / \ 0 / \ . o
)(t,xt ) > V(t)i(t,xt ), for i G N and t G [t0,T], yields a time consistent solution to the cooperative game r(x0,T — t0).
Proof. Note that any solution optimality principle as that in Theorem 3.1 satisfies
(i) group optimality, (ii) individual rationality. Moreover, with a0 being chosen in
all the subgames P(xT , T — t) for t G [t0, T], we have
(tn)(t,xT ) exp /tTn r(y)dy = (t)(t,xt ) = (t)(t,xt )
= /TT gi[s,xTt> (s)^ (tn )T (s,xT0 (s)),^(tn )T (s,xT0 (s)), ••• ,^itn )T (s,xTt> (s))]
x exp [— JT r(y)dy] ds + exp — JtT r(y)dy q%(xT (T)), for i G N.
Hence (3.12) is satisfied. The solution optimality principle given in Theorem 3.1 is indeed time consistent. □
Yeung and Petrostyan (2006a) have provided a method to derive individual cooperative payoffs in an analytically tractable form. Specific solution optimality principles leading to time consistent solutions in a few classes of cooperative differential games with nontransferable payoffs can be found in Yeung and Petrosyan (2006a). The deterministic versions of Yeung and Petrosyan (2006a) and Yeung et al (2006) are also examples of nontransferable payoff differential games with time-consistent solutions.
3.3. Subgame Consistency Solutions in Cooperative Stochastic Differential Games
As one can imagine the derivation of tractable solutions in cooperative stochastic differential games could be rather difficult because of the games’ complexity. Haurie et al (1994) derived cooperative equilibria in a stochastic differential game of fishery with the use of monitoring and memory strategies. In the presence of stochastic elements, a more stringent condition - subgame consistency - is required for a dynamically stable cooperative solution.
The notion of subgame consistency in cooperative differential games was first examined by Yeung and Petrosyan (2004). In particular, a cooperative solution is subgame-consistent if an extension of the solution policy to a subgame at a later
starting time and any feasible state brought about by prior optimal behavior would remain optimal. Conditions ensuring subgame consistency in cooperative solutions of stochastic differential games generally are more analytically intractable than those ensuring time consistency in cooperative solutions of differential games.
Let rs (x0, T — t0) denote the cooperative stochastic differential games (2.15)-
(2.16) with transferable payoffs. To achieve group optimality the players would seek to maximize the expected joint payoff
{T n s
/to £ [s,x(s),ui(s),u2(s),-,un(s)] exp[— /t0 r(y)dy]
j=1 n , (3.13)
+ E exp[ /toT r(y)dy] (x(T))
j=i
subject to (2.15).
Let j [^1*0 ^ (t,x),^0 ^ (t,x), ••• ,^nt0 ^ (t,x),/ort € [t0,T ] j, denote a set of
controls (if exists) that provides an optimal solution to the stochastic control problem (3.13)-(2.15).
Substituting this set of control into (2.15) and solving yields the optimal (cooperative) trajectory in the form of a stochastic path {x * (i))T=t0 where
ft * * c*(i) = x0 + / / [s,x* (s),^1i0} (s,x* (s)),^2t0} (s,x* (s)), ••• ,^nt0 }* (s,x* (s))]ds
n ^ ¿0
+ i <r[s,x*(s)]dz(s). (3.14)
J ¿0
We use X* to denote the set of realizable values of x * (t) at time t generated by
(3.14). The term x * is used to denote an element belonging to the set X*. We use rs (x*, T — t) to denote the cooperative stochastic differential game with payoffs
(2.16) and dynamics (2.15) starting at time t given the state at time t is x*€ X*.
In the start of the game, the players agree to adopt a solution optimality principle 0(x* ,T — t) which states that at current time t with current state x* the players would maximize the expected joint cooperative payoff and allocate to the player i a cooperative payoffs over the period [t,T] equaling
£(t)l(t, x*), for i € N and t € [t0,T]. (3.15)
The vectors £(t^(t,x*) = [£(t)1(t,x*),£(t)2(t, x*),••• ,£(t)n(T,x*)], for t € [t0,T], are valid imputations if the following conditions are satisfied.
Condition 3.2
(i) £(t}( t, x*), for t € [t0 ,T] and x*€ X*, is a Pareto optimal imputation vector,
(ii) £(t(t,x*)> V(t^(t,x*), for i € N, for t € [t0,T] and x*€ XT,
where V(t(t, x*) is the payoff to player i in the noncooperative version of the game Ts (x*, T — t). □
In particular, part (i) of Condition 3.2 ensures Pareto optimality, while part
(ii) guarantees individual rationality.
Once again, one can readily observe that the solution optimality principle 0(x*, T — t) would remain in effect only it assigns an imputation vector £(t)(t,x* ) = [£(t)1(t, x* ),£(t)2(t,x* ), ••• ,£(T)n(T,x*)] to the subgame r(x*,T — t) satisfies the condition that
£(t0 )l (t, x* ) exp
'’(y)dy
= £(T }í(t,x;),
(3.16)
for i € N and t € [to , T].
T
4. Payoff Distribution Procedures Leading to Dynamically Stable Solutions
In this section, we consider payoff distribution procedures leading the dynamically consistent solutions.
4.1. Payoff Distribution Procedures for Cooperative Differential Games
A payoff distribution procedure (PDP) for cooperative differential games (as proposed in Petrosyan (1997) and Yeung and Petrosyan (2006a)) must be now formulated so that the agreed-upon imputations can be realized. For the condition in (3.7) to hold, it is required that BT(s) = B|(s), for i G [1,2] and t G [t0,T] and t G [to,T] and t = t. Adopting the notation BT(s) = Bt(s) = B^(s) and applying Definition 5.1, the PDP of the subgame consistent imputation vectors £(t) (t, xT) has to satisfy the following condition.
Condition 4-1 The PDP with B(s) and q(x*(T)) corresponding to the time consistent imputation vectors £(t)(t, x*) must satisfy the following conditions:
(i) £ (s) = £ [s,x*,^(t) (s,xS(s,xS)], for s € [t0,T];
¿=1 j=i
(ii) j T Bi(s)exp [— j rs r(y)dy] ds+qi(x * (T ))exp — / J r(y)dy > V (t)^(t,x* ), for i € {1,2} and t € [t0,T]; and
(iii) £(T)i(T,x*)=jT+4i Bi(s)exp [— jrs r(y)dy] ds+exp — j ^+At r(y)dy
x£(T+4t)i(t + 4t,x* + 4x*), for t € [t0,T] and i € {1,2}; where
4x* = /[t,x*,^(t) (t,x*,^(t) (t,x*)]^t + o(^t),
□
Consider the following condition concerning £(T)(t,x *), for t € [t0,T] and t € [t,T]:
Condition 4.2 For i € {1,2} and t > t and t € [t0,T], the terms £(T)i(t,x*) are functions that are continuously twice differentiable in t and xt . □
If the imputations £(t) (t,x *), for t € [t0 ,T], satisfy Condition 4.2, one can obtain the following relationship:
/T Bi(s)exp [—/Tr(y)dyds
= £(T)i(t, x*) — exp — JTT+^i r(y)dy £(T+^t)i(t + ^t,x* + 4x*)
= £ (T)i(T,x*) — £ (T)i(T + ¿t,x* + 4x* ),
for all t G [to,T] and i G {1,2} .
(4.1)
With Æ ^ 0, condition (4.1) can be expressed as:
Bi (t )^t = —
£(T)i(t,x* )
x/[t,x*,^(t) (t,x*),^^ (t,x*)]^t — o(^t).
£X*)i(t,x * )
(4.2)
(t)*
Dividing (4.2) throughout by 4t, with 4t ^ 0, yield
Bi(T) = — [£i(T)i(t,x* )|t=T] — [£X** (t,x* )|t=T]/[t,x^^^ (t,x* (t,x* )].
î(t)*
(t)*
/,(t)*
(4.3)
Therefore, one can establish the following theorem.
Theorem 4.1. If the solution imputations £(t)*(t,x*), for i G {1,2} and t G [to,T], satisfy Condition 4-1 and Condition 4-2, a PDP with a terminal payment q*(xT)) at time T and an instantaneous payment at time t G [to,T]:
b»(t) = — [£t(T)i(t,x*)|t=T] - [£Xf(t,x*)|t=T]/[t,<,^(t) (t,<),^(t) (t,<)], for i G {1, 2},
yields a time consistent solution to the cooperative game rc(xo ,T — to).
t=T
t=T
4.2. PDP under Specific Optimality Principles
Consider a cooperative game rc(x0,T — t0) in which the players agree to maximize the sum of their payoffs and divide the total cooperative payoff equally. The imputation scheme has to satisfy:
Proposition 4.1. In the game Tc(x0,T — t0), an imputation
£^(i0,xo) =V^'>i(t0,x0) + -
W(t0)(to, xo) — ^ V(t0)j (to, xo) j=i
is assigned, to player i, for i G {1, 2};
and in the subgame rc(x*,T — t), for t G (to,T], an imputation
£(T)i(T,0 = V^i(r,x*) + -
is assigned, to player i, for i G {1, 2}.
W(t)(t,x* ) V(T)j (t,x* )
j=1
□
One can readily verify that £(t satisfies Conditions 4.1 and 4.2.
Using Theorem 4.1 a PDP with a terminal payment q% (x(T)) at time T and an instantaneous imputation rate at time t G [t0,T]:
Bi(T) =
Vt(T)i(i,xt )
Wt(T)(i,xt )
Vt(T)j (t,xt)
+
+
+
Vx(tT )l(i,xt)
f [r,xT,^jT) (t,xt ),^(T) (t,xt )]
WÍT)(í,xt) f [t,xt,^1T J (t,xt ),^^ (t,xt )]
( )
( )
Vx(tT )j (t,xt ) f [t,xt,^^ (t,xt ),^^ (t,xt )]
( )
( )
(4.4)
for i, j G {1, 2} and i = j, yields a time consistent solution to the cooperative game rc(xo,T — to), in which the players agree to divide their cooperative gains according to Proposition 4.1.
4.3. Payoff Distribution Procedures for Cooperative Stochastic Differential Games
The PDP with B(s) and q(x(T)) corresponding to the subgame consistent imputation vectors £(t)(t, xT) must satisfy the following conditions:
Condition 4-3
2 2
(i) E B¿(s)=J] [s,xs,^ (t) (s,xs),^2T; (s,xs)], for s G [íq,T];
( )
j=1
j=1 ■T
x(t )
(ii) Et /TB¿ (s)exp [-/TSr(y)dy] ds+q¿(x(T))exp -/t r(y)dy xT} > V(t)í(t,xt), for i G [1,2] and t G [to,T]; and
(iii) £(t)í(t,xt)=Et I ^ /T+Zií Bi(s) exp [-/rs r(y)dy] ds+exp - /TT+Æ r(y)dy e(T+4i)i(T + 4i,xT + A&,-) ^
x
where
x(t) = xT >, for t G [to,T] and i G [1,2];
4xT = f [t, xT, ^(t) **(t, xT, (t, xT)]^t + ff[r, xT]^zT + o(^i),
x(t) = xT G Xrio) , 4zT = z(t + Æ) — z(t), and ET [o(Æ)]/Æ ^ 0 as
Æ ^ 0. □
Consider the following condition concerning subgame consistent imputations C(t)(t,xt), for t G [to,T]:
( )
1
t=T-
t=T-
t=T-
t=T-
Condition 4-4 For i G [1,2] and t > t and t G [to,T], the terms £(T)i(t,xt) are functions that are continuously twice differentiable in t and xt. □
If the subgame consistent imputations £(t) (t, xT), for t G [t0 , T], satisfy Condition 4.4, a PDP with B(s) and q(x(T)) will yield the relationship:
Et
Bi (s) exp
t ^ T
Et I £(r)i(t,Xt) - exp - J
r(y)dy
ds
x(t ) = xr
r+^t
r(y)dy x(t ) = xT
£ (r+Æ)i(t + 4i,Xr + ^Xr)
Er j e(r)i(t,Xt) - e (r)i(t + 4i,Xr + ^Xr) x(t)= Xr j , (4.5)
or all t G [t0, T] and i G [1, 2].
With Æ ^ 0, equation (4.5) can be expressed as: Et {Bj (t)Æ + o(Æ)} =
eXl)i(i,Xt) / [T,Xr,^ir) (T,Xr),^2r) (t,xt)]^Î
i=r.
Æ -
eXl)i(t,Xt) CT[t,xt]^zr - 0(^i)
(4.6)
Taking expectation and dividing (4.6) throughout by with ^ 0, yield
Bi(T ) =
ei(r)i(t,Xt)
J2 fihC(r,xT) h,C=l
eXl)i(i,Xt) f [t,xt,^^ (t,xt),^^ (t,xt)]
( )
''r;, </,2T)
(4.7)
Therefore, one can establish the following theorem.
Theorem 4.2 (Yeung and Petrosyan (2004)). If the solution imputations £(t)*(t;xt), for i G [1, 2] and t G [to,T], satisfy Conditions 4-3 and 4-4, a PDP with a terminal payment ql(x(T)) at time T and an instantaneous imputation rate at time t G [to,T]:
Bi(T ) = - ei(r)i(t,Xt)
eXl)i(i,Xt) f [t,xt,^^ (t,xt),^^ (t,xt)]
h,Z=1
for i G [1, 2],
yields a subgame consistent solution to the cooperative game rc(xo, T — to).
□
s
i=r
i=r
i=r
t=r-
t=r
t
5. Noncooperative Equivalence Imputation and Mechanism Design
In this section, we examine noncooperative equivalence imputation as a benchmark of allocation in optimal resource use mechanism design. Consider a dynamic market situation in which there are n economic agents with initial state x0 and duration T — to. The state space of the game is X G Rm, with permissible state trajectories citation {x(s),to < s < T}. The state dynamics of the game is characterized by the vector-valued differential equations:
X(s) = f [s,x(s),ui(s),u2(s), ••• , un(s)], x(to) = xo, (5.1)
where Mj(s) G Rmi is the control vector of agent i.
The objective of agent i is
/t0 gi[s,x(s),ui(s), u2(s), ••• ,u„(s)] ds + «¿M^X (52)
for i G {1, 2, ••• , n} = N,
and gi [s,x(s),M1(s),u2(s), ••• ,un(s)] and ql(x(T ))are non-negative. The agents’ payoffs are transferable.
Invoking the work of Isaacs (1965) and Bellman (1957) a feedback Nash equilibrium of the game can be characterized the following well-known theorem:
Theorem 5.1. An n-tuple of strategies {^(t,x), for i G N} provides a feedback Nash equilibrium solution to the game (5.1)-(5.2) if there exist continuously differentiable functions Vi(t,x) : [to,T] x Rm ^ R, i G N, satisfying the following set of partial differential equations:
-V/ (i,x) =
max {gi [t X ^i (t x) , ^2 (t x) ,''' , ^¿-1(t x) , , ^¿+i(^ x) ,''' , (t x)]
Ui 1
+ VX (t x)f [t X, ^1 (t x) , ^2 (^ x) , ' ' ' , ^¿-1 (t x) , , ^¿+1 (^ x) , ' ' ' , ^n (t x)] }
V¿(T, x) = (x), i G N.
The noncooperative payoff of agent i at time t given that x(t) = t is given by the continuously differentiable function V¿(t,x).
Now consider the attempt to bring about group optimality. To achieve group optimality one has to maximize the agents’ joint payoff:
/T n n
53gj [s,x(s),u1(s),u2(sX ••• ,un(s)] ds + 53 (x(t )) (5.3)
io j=1 j=1
subject to (5.1).
Let {^¿(s,x), for i G N} denote a set of strategies leading to an optimal control solution of the problem (5.1) and (5.3) the total payoff under the group optimal scheme over the interval [t, T] where t G [to, T] is:
/T n
53 gj [s,X * (s),^1(s,x * (s)),^2(s,x * (s)) ■ ,^n(s,x *(s))] ds+
; j=1
n
+ £ (x* (T)). (5.4)
j = 1
The state dynamics under cooperation is: x *(s) = f [s,x * (s),^i (s,x * (s)),^2 (s,x * (s)), ••• ,^n (s,x * (s))], x(to) = xo. (5.5) The corresponding optimal trajectory can be expressed as:
x* (t) = xo + i f [s,x* (s),^*(s,x* (s)),^* (s,x* (s)), ••• ,^n (s,x* (s))] ds. (5.6)
j to
For notational convenience, we use the terms x * (t) and x * interchangeably.
5.1. Noncooperative-Equivalent Imputation Formula
The minimum requirement to induce the agents to adopt the optimal strategies (s,x), for i € N} is that each agent would receive a payoff at least be equal to his noncooperative payoff. Moreover, to maintain individual rationality throughout the duration T-to, each agent’s imputation must at least be equal to his noncooperative payoff at each time instant t € [t0, T] along the optimal state path {x * (t)}.
Let f (t, x*) denote the imputation to agent i under the group optimal scheme over the time interval [t,T] along the optimal path {x*}T=to for t € [t0, T]. An imputation distribution procedure as in Petrosyan and Danilov (1982) and Yeung and Petrosyan (2004 and 2006a) has to be formulated so that the cooperative imputation (t, x*) = Vj(t, x*) can be realized along the optimal path. To do this we let Bi(s,x *(s)) denote the instantaneous rate of payment received by player i at time s. In particular,
(t,x* ) = Vj (t,x* )=y B (s,x * (s)) ds + cl (xT), for t € [to,T ]. (5.7)
Theorem 5.2. A payment scheme with a terminal payment ql(xT) at time T and an instantaneous rate of payment at time t € [to,T ] along the cooperative trajectory
{x* }T=to being
Bi (t,x* ) =
- Vr ^ x*) - VX * (T x*)f [т, x* , ^1 (т, x*) , ^2 x*) , ■ ■ ■ , ^n (T x*)],
(5.8)
yield the noncooperative-equivalent imputation
f (t,x*) =J T Bi(s,x* (s)) ds+ql (xT)= V*(t, x*), for t € [to ,T]. □
Proof. Using (5.7) one can obtain the identity:
i (s,x *(s)) ds + V1(t,x*) = V1 (to,xo). for t € [to,T]. (5.9)
to
Differentiating (5.8) with respect to t yields
B»(t) = -dV' (t,x* )/dT = -Vr (t,x* ) - VX* (t,x* )x*(t) (5.10)
Invoking (5.6) we obtain
Bi(T) = -Vr(T,x*) - Vx\ (T,x*)f [t,x*,^1(t,x* ),^2 (T,x*), ,^n(T,x*)]. (5.11)
Hence Theorem 3.1 follows. □
Theorem 5.1 yields a distribution formula for noncooperative-equivalent imputation in an group optimal scheme. Such a formula can be obtained in closed form for any explicitly solvable games.
5.2. An Economic Exegesis of the Formula
An economic exegesis of the rationale for the noncooperative-equivalent imputation formula (5.11) can be obtained. Note that the Isaacs-Bellman equation in Theorem 5.1 for a feedback Nash equilibrium in the noncooperative game (5.1) and (5.2) leads to
-Vt(t,x* ) = gi[T,x*,^1(T,x* ),^2(t,x* X ,^n(T,x* )]
(5.12)
+VX* (T,x* )f [t,x*A(t,< X^2 (T,x* ), ^n(T,x* )], for T € [to,T ].
Using (5.12) the distribution formula in (5.10) can be expressed as:
Bi(T,x* ) = gi[T,x*,^1 (t,x* X^2(t,x* ), ,^n(T,x* )]
+ VX*(T,x*) {f [t,x*A(t,< ),^2(T,x* X ,^n(T,x* )] (5.13)
- f [t,<^1(t,< ),^2(t,< ), ,^n(T,x* )]} .
However, along the optimal path {x* }T=to, the instantaneous rate of payoff to agent i is
gj[T, x*,^1(t, x* ),^2 (t, x*), ••• ,^n(T,x*)] at time instant t. (5.14)
In order for agent i to realize an instantaneous rate of payoff equaling B^t, x*) a noncooperative-equivalent compensation formula can be obtained as
(T,x*) = Bi(T,x*) - gi[T,x*,^1(T,x* ),^2(T,x*), ,^n(T,x* )],
or:
^i (T,x* ) = gi[T,x*,^1(T,x* X^2 (T,x* ), ••• ,^n(T,x* )]
- g [т, x*, ^1 (т, x*) , ^2 ^ x*) , ■ ■ ■ , ^n (T x*)]
(5.15)
+ VX* (т, x* ) {f [т, x*, 4>1 (т, x* ) , ^2 x* ) , ' ' ' , ^n (T x* )]
- f [t,x*A(t,< ),^2(t,x* ), ,^n(T,x* )]} .
In formula (5.15) the term
g K x*, ^1 (т, x*) , ^2 ^ x*) , ' ' ' , ^n ^ x*)]
-g [т, x*, ^1 (т, x*) , ^2 ^ x*) , ' ' ' , ^n (T x*)]
yields the difference between agent i’s rate of instantaneous payoffs when he uses the noncooperative strategy and that when he adopts the group optimal strategy. The term VX * (t, x*) reflects the marginal effects of a change in the state variables on agent i’s noncooperative payoff. The term f [t, x*, ^1 (t, x*), ^2(t, x*), ■ ■ ■ , ^n(T, x*)]
yields the instantaneous change of the states over time if the agents act noncooper-atively, while the term f [t, x*,^1 (t,x*),^2(t,x*), ■ ■ ■ ,^n(T, x*)] yields the instantaneous change of the states over time if the agents act group optimally.
Hence, the expression
VX* x*) {f [т, x^ ^1 (т, x*) , ^2 (т, x*X ••• , ^n(т, x*)]
-f K x*, ^1 x* ), ^2 x*) , , ^nx*)]}
represents the compensation to agent i when the change in the state variable follows the group optimal trajectory instead of the noncooperative path.
To sum up, at time instant t the compensation to agent i leading to the noncooperative-equivalent instantaneous rate of payoff Bi (t, x*) consists of
(i) the compensation on the difference between agent i’s rate of instantaneous payoffs when he uses the noncooperative strategy and that when he adopts the group optimal strategy, and
(ii) the compensation to agent i for the difference in the change in the state variable on the group optimal trajectory and that on the noncooperative path.
nn
Finally if a payment A, where E A< W(to,xo)- E Vj(to,xo), is given to
j=1 j=1
agent i <G N at terminal time T and all the players are willing to adopt the group optimal strategies, optimal resource use will result and all agents and the planning body (government) will be better off under the group optimal scheme.
6. Mechanism Design for Global Environmental Management
After several decades of rapid technological advancement and economic growth, alarming levels of pollutions and environmental degradation are emerging all over the world. Though cooperation in environmental control holds out the best promise of effective action, limited success has been observed. Existing multinational joint initiatives like the Kyoto Protocol can hardly be expected to offer a long-term solution because (i) the plans are limited to a confined set of controls like gas emissions and permits which is unlikely be able to offer an effective mean to reverse the accelerating trend of environmental deterioration, and (ii) there is no guarantee that participants will always be better off and hence be committed within the entire duration of the agreement.
To create a cooperative solution a comprehensive set of environmental policy instruments including taxes, subsidies, technology choices, pollution abatement activities, pollution legislations and green technology R&D has to be taken into consideration. The implementation of such a scheme would inevitably bring about different implications in cost and benefit to each of the participating nations. To construct a cooperative solution that every party would commit to from beginning to end, the proposed arrangement must guarantee that every participant will be better-off and the originally agreed upon arrangement remain effective at any time within the cooperative period for any feasible state brought about by prior optimal behavior.
6.1. An Analytical Framework
Consider a global economy which is comprised of n nations. At time instant s the demand system of the outputs of the nations is
Pi (s) = fj Ms),q2 (s), ,qn(s),s], i G N = {1, 2, ••• ,n}, (6.1)
where Pi (s) is the price vector of the output vector of nation i and q? (s)is the output
of nation j. The demand system (6.1) shows that the world economy is a form of generalized differentiated products oligopoly.
Industrial profits of nation i at time s can be expressed as:
fj[q1 (s),q2(s), ••• ,qn(s),s]qj (s) - cj [qj (s),v (s)], for i G N, (6.2)
where vi (s) is the set of environmental policy instruments of government i and cj[qi(s),vi(s)] is the cost of producing qi(s) under policy vi(s).
Profit maximization by the industrial sectors yields a market equilibrium in which nation i’s instantaneous output as:
q* (s) = qj [v1 (s),V2 (s), ••• ,vn (s),s] = qj [v(s),s], for i G N. (6.3)
One can readily observe from (6.3) that each nation’s output decision depends on government environmental policies.
Let x(s) C Rm denote the level of pollution at time s, the dynamics of pollution stock is governed by the stochastic differential equation:
dx(s) =
£"=i q (s),vj(s)] - E n=i K' (s)>x(s)]- á [x(s)]x(s)
ds+
+ct [x(s)]dz(s), (6.4)
x(to) = xto,
where a is a noise parameter and z(s) is a Wiener process,
a?[qj(s),v?(s)] is the amount of pollution created by q?(s) amount of output produced under policy vi (s),
u? (s) is the pollution abatement effort of nation j,
b? [u?(s),x(s)] is the amount of pollution removed by u?(s) unit of abatement effort of nation j, and ¿[x(s)] is the natural rate of decay of the pollutants.
The governments have to promote business interests and at the same time handle the financing of the costs brought about by pollution. In particular, each government maximizes the gains in the industrial sector plus tax revenue minus expenditures on pollution abatement and damages from pollution. The instantaneous objective of government i at time s can be expressed as:
/ ' [qi(s),q2(s), ••• ,q„ (s),s]qi (s) - cl [qi (s),v¿ (s)]
-cf [vi(s)] - c“[ui(s)] - [x(s)], * G N,
(6.5)
where cP [vi (s)] is the cost of implementing the vector policy instrument vi (s), c“[ui(s)] is the cost of employing uiamount of pollution abatement effort, and hi [x(s)] is the value of damage to country i from x(s) amount of pollution.
The governments’ planning horizon is [t0,T]• It is possible that T may be very large. The discount rate is r. At time T, the terminal appraisal of pollution damage is $l[x(T)] where dgl/dx < 0.
Substitute ^¿(s), for i € N, from (6.3) into (6.4) and (6.5) one obtains a stochastic differential game in which government i € N seeks to:
max E,
(s)
fHlT[v(s), s], ç2[v(s),s], ■■■ ,qn[v(s),s], s}<f[v(s), s]
-ci{qi[v(s),s],vi(s)}- cf [vi(s)] - c“[ui(s)] - hi[x(s)] +gi[x(T )]e-r(T-to)
e-r(s-t0)ds
(6.6)
subject to
dx(s) = ZjLi [v(s),s],Vj(s)}^£n=i bj[uj(s),x(s)] - 6[x(s)] x(s) ds+
+ct[ x(s)]dz(s), x(to) = xto,
(6.7)
Explicit illustrative examples of the above theoretical framework can be found in Yeung (2007) and Yeung and Petrosyan (2008).
6.2. International Cooperation in Environmental Control
Now consider the case when all the nations want to cooperate and agree to act so that an international optimum could be achieved. Since nations are asymmetric and the number of nations may be large, a reasonable solution optimality principle for gain distribution is to share the expected gain from cooperation proportional to the nations’ relative sizes of expected noncooperative payoffs. Hence the solution imputation scheme (r)i(r, x*); for i € N} has to satisfy:
Condition 6.1.
e(r)i (r,x; ) = v(T)i (r,x; ) +
V(t^(t, X*)
n
Y v(T)j (t,x; ) j = l
w(T)(r,x;) -V v(T)j (r,x;)
j=l
n
Y V(T)j (t,x; ) j=l
-w (t)(t, x; ),
(6.8)
for i € N, x; G X; and t € [t0,T].
□
A distribution scheme with a terminal payment —gl[x^ — xl] at time T and an instantaneous payment at time t € [t0,T]:
£ «j(i5'[^(t,<),tL^(t,x*)}
j=1
(6.9)
- E bj K- (T,x*X x*] - á(<) < j=i
for i € N,
yield Condition 6.1.
6.3. Policy Design
Using the above analysis as a policy guide, a grand coalition of all nations should be formed to pursue a comprehensive cooperative scheme of industrial pollution abatement. In particular, the entire set of policy instruments available - including environmental taxes and charges, adoption of environment-friendly production technology, subsidy to the replacement of polluting techniques, joint research and development in clean technology, restoration and preservation of the natural ecosystem, and legislations to outlaw environmentally unacceptable practices - will be used achieve an optimal cooperative outcome. A payment distribution mechanism has to be formulated so that cooperative gains will be shared according to the proportions of the nations’ relative sizes of expected noncooperative payoffs throughout the planning horizon. Appropriate policy coordination will lead to the enhancement of economic performance and the realization of a cleaner environment.
A particularly relevant mechanism design would be the formation of a United Nations Agency to coordinate international cooperative actions on pollution and climate change. The Agency is would be comprised of three divisions. An executive branch would be established to coordinate adoption and development of clean technology, pollution abatement activities, use of materials, waste disposal, mode of resource extraction and cooperation in environmental R&D. A financial branch (or FUND) would be set up to handle pollution charges, clean technology subsidies, financial aids and payoff distributions so that the agreed upon optimality principle will be realized throughout the cooperative period. Lastly, a legislative body would be in place to enact regulations on the use of dirty technologies, toxic disposal, pollutant emissions, activities damaging the environment and violation of the cooperative agreement.
t=T
t=T
7. Financial Constraint and Irrational-Behavior-Proof
In this section we explore the effects of financial constraint and irrational behavior on mechanism design involving cooperative differential games.
7.1. Financial Constraint and Mechanism Design
Consider the n—person nonzero-sum differential game with initial state x0 and duration T —10. The state space of the game is X € Rm, with permissible state trajectories citation{x(s), to < s < T}. The state dynamics of the game is characterized by the vector-valued differential equations:
X(s) = f [s,x(s),ui(s),u2(s), ■ ■ ■ , un(s)], x(to) = xo,
(7.1)
where u(s) € is the control vector of player i.
The objective of player i is
r T
/ gj[s,x(s),ui (s),«2(s), ■ ■ ■ , un(s)] e-r(s-to)ds + e-r(T-to)qj(x(T)),
j to
for i € {1, 2, ■ ■ ■ , n} = N
Let {^j(s,x), for i € N} denote a set of strategies leading to a feedback Nash
equilibrium, the game equilibrium strategies can be obtained as:
x(s) = f {s,x(s),^i [s,x(s)],^2 [s, x(s)], ■ ■ ■ ,^„[s, x(s)]}, x(to) = xo. (7.2)
We denote the solution to (7.2) by {x(s) }T=to, and use the terms x(s) and
xsinterchangeably. The noncooperative payoff of player i over the interval [t,T] where t € [to,T] is:
(t,xt) = 11 gj [s,x(s),^i (s,x(s)),^2(s,x(s)), ' ,^n(s,x(s))] e-r(s-to} ds
+e-r(T-to) qj (x(T)),
(7.3)
for i € {1, 2, ■ ■ ■ , n} = N, and x(t) = xt.
Under cooperation group rationality required the players to maximize their joint payoff
/T n n
[s,x(s),ui (s), U2(s), ■ ■ ■ ,«n(s)]e-r(s-to} ds ^53 e-r(T-to}(x(T))
;° j=i j=i
(7.4)
subject to (7.1).
Let {^j(s,x), for i € N} denote a set of strategies leading to an optimal control solution of the problem (7.1) and (7.3) and let {x * (s) }T=to denote the optimal cooperative path, the total cooperative payoff over the interval [t, T] where t € [to, T] is:
rp n
W (t,x*) = / tZ gj [s,x * (s),^i(s,x * (s)),^2(s,x *(s)), ••• ,^n(s,x *(s))]
j=i n (7.5)
xe-r(s-to)ds + ^ e-r(T-to)qj(x *(T)). j = i
Let (t, x*)=/ T Bj(s) e-r(s-T)ds+qj (xT) denote the imputation to player i under
cooperation over the time interval [t, T] along the cooperative path {x*}T=to for t € [to, T].
Since an imputation satisfies group and individual rationalities, we have:
(i) W (t, x*) = £ (t, x*), and
j=i
(ii) f(t,x*)> V¿(t,x*), for i € N.
In a noncooperative equilibrium, the payoff received by player i in the interval [to, t] can be expressed as:
/ to gj [s,x(s),^i (s,x(s)),^2(s,x(s)), ••• ,^n(s,x(s))] e-r(s-to )ds
= Vj (to ,xo) — Vj (t,xt). .
The cooperative payoff received by player i in the interval [to, t] can be expressed as:
f(T,< ) = / "Bi (s) e-r(s-to)ds = ei(io,Xo) - f (t, x;). (7.7)
to
If player i’s cooperative payoff in the in the interval [t0 ,t] is smaller than his noncooperative payoff in the in the interval [t0 ,t] and he cannot finance the deficit (to,x0)—£*(t,x;) —[V1 (to,x0) —Vi(t,xT)], he would reject the optimality principle leading to the imputation £*(t, x;). Therefore, financial constraints often posted a problem in cooperation.
Consider the analysis in Section 6, sharing the expected gain from cooperation proportional to the nations’ relative sizes of expected noncooperative payoffs is reasonable and acceptable to a large number of asymmetric nations. However, the failure of some (developing) nations to finance the deficit
f (to,xo) — f (t,x; ) — [Vl(to,xo) — V1 (t,xt)]
may create severe strain on the cooperative scheme. Often these nations would request to be exempted from carrying out the optimal strategies (as in the case of the Kyoto Protocol). This is certainly a suboptimal arrangement and could reduce the gain from cooperation substantially. As proposed before, financial aid should be given to these participants (with repayment made later) so that they can carry out the optimal strategies.
7.2. Irrational-Behavior-Proof Condition
We have shown that given subgame-consistent imputations satisfy group and individual rationalities throughout the cooperative trajectory, no rational players will deviate from the cooperative path. However, in reality irrational behavior may appear for various reasons. For instance, a player may use ‘irrational’ acts to extort additional gains if later circumstances allow. Refusal of other players to yield to his extortion would result in the dissolution of the cooperative scheme.
Once again an imputation satisfies group and individual rationalities, and therefore:
(i) W(t,x;)=£ (t,x;), and
j=i
(ii) f (t, x; )> Vi (t, x;), for i e N.
Moreover, one can write
r T
f (t,x;) = J Bi(s) e-r(s-to)ds + e-r(T-to)qi(xT), for i e N. (7.8)
Consider the case where the cooperative scheme has proceeded up to time t and some players behave irrationally leading to the dissolution of the scheme.
At time t if the condition
i Bi(s) e-r(s-to)ds + Vi(t,x;) > Vi(to,xo) (7.9)
to
is satisfied player i is irrational-behavior-proof (I-B-P) because irrational actions leading to the dissolution of cooperative scheme will not bring his resultant payoff
below his initial noncooperative payoff. On the other hand, if
i B(s) e-r(s-to)ds + Vi(r,x;) <V^(to,x0), (7.10)
j to
player i is not proofed from irrational-behavior-proof of other player.
To check whether the irrational-behavior-proof condition holds, one can invoke (7.8) and express the I-B-P condition in (7.9) as:
f(io,x0) - f (t,x*) + ) - V^(io,x0) > 0, for i G N. (7.11)
In an explicitly solvable game, given a time-consistent and hence continuously differentiable imputation £*(t, x*) one can verify whether the I-B-P condition is satisfied along the cooperative path {x*}T=to. In particular, the I-B-P condition may or may not be satisfied throughout the game horizon. Given in exact analytical form, the satisfaction of it along the cooperative trajectory could be obtained for explicitly solvable cooperative differential game with time-consistent and differentiable imputations (like those in Jrgensen and Zaccour (2001) and Yeung and Petrosyan (2004, 2006a)).
7.3. Properties of the I-B-P Condition
Several interesting properties of the I-B-P condition worth noting.
Property 1. At terminal time T and at initial time t0 the I-B-P condition holds.
Proof. At time T the I-B-P condition becomes:
f (t0,X0) - f (T,xT) + V'(T,xT) - V*(t0,X0) > 0. (7.12)
From (2.3), we have Vl(T,x^)=ql(xT) and from (7.8) we have (T,xT)=ql(xT).
Substituting these into (7.12) yields £l(i0,x0)-V1 (t0,x0) > 0 and therefore the I-B-P condition holds.
At initial time t0,
f (t0,x0) - f (t0,X0) + Vl(i0,x0) - Vl(i0,X0) = 0. (7.13)
Hence the I-B-P condition again holds. □
Property 2. There exists a time interval [ij,T] for player i G N such that the I-B-P condition holds.
Proof. This is a direct result of Property 1. □
Property 3. A sufficient condition for the I-B-P condition to hold for player i
throughout the game interval [t0,T] is that his instantaneous rate of cooperative
payment
Bj(t) > -dVl(t,x*)/dT at all t G [t0,T] (7.14)
along the cooperative path.
Proof. Recall the I-B-P condition in (7.9)
i B(s) e-r(s-to)ds + V*(t, x*) > Vi(t0,x0). (7.15)
to
Note that the equality sign in (7.16) will hold at t = t0. If the time derivative on the left-hand-side of inequality (7.16) exceeds that on the right-hand-side all the time, that is
Bi(t) e-r(T-to) + dVl(T,x*)/dT > 0, for all t G [t0,T], (7.16)
the I-B-P condition of player iwill hold throughout the game interval [t0,T]. Hence Property 3 follows. □
Finally, the condition can also be applied to cooperative differential games with nontransferable payoffs. Let £(t, x^) denote the imputation to player i under cooperation over the time interval [t, T] along the cooperative path {x^}T=to where a is the agreed-upon cooperative weights. Using Yeung’s (2004) technique to obtain (t, x^) one can arrive at an I-B-P condition for nontranferable payoffs game as:
£(a)i(t0,x0) - £(a)i(t,x?) + V^x?) - V*(t0,x0) > 0, for i G N. (7.17)
One can readily show that properties similar to Property 1 and Property 2 hold. The satisfaction of condition (7.17) along the cooperative trajectory can be verified for explicitly solvable cooperative differential game with nontransferable payoffs (like those in Yeung and Petrosyan (2005)). Markovkin (2006) analyzed the I-B-P condition in linear quadratic differential games.
8. Concluding Remarks
Economic analysis no longer treats the economic system as given since the appearance of Leonid Hurwicz’s pioneering work on mechanism design. The design point of view enlarges our vision and helps economics avoid a narrow focus on existing institutions. The failure of the market to provide an effective mechanism for optimal resource use will arise if there exist imperfect market structure, externalities, imperfect information or public goods. These phenomena are prevalent in the current global economy. As a result not only inefficient outcomes like over-extraction of natural resources had appeared but gravely detrimental events like catastrophe-bound industrial pollution had also emerged under the conventional market system.
Cooperative games suggest the possibility of socially optimal and group efficient solutions to decision problems involving strategic actions. This lecture focuses on cooperative game-theoretic design of mechanisms for optimal resource use. Since resource use is often a dynamic process we concentrate on mechanism design involving an intertemporal framework. Crucial features that are essential for a successful mechanism - individual rationality, group optimality, dynamic consistency, distribution procedures, budget balance, financing, incentives to cooperate and practicable institutional arrangements - are considered. Cooperative game-theoretic mechanism design is used to establish the foundation for an effective policy menu to tackle suboptimal resource use problems which the conventional market mechanism fails to resolve.
Large scale design of mechanisms is in order for research in optimal resource use involving cooperative dynamic game theory. This analysis is expected to open up a policy forum for the design of new economic institutions and contribute new directions of research in the field of neo-institutional economics.
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