Contracting theory with competitive interacting agents
aa r X i v : . [ q -f i n . E C ] M a y Contracting theory with competitive interacting Agents
Romuald
Elie ∗ Dylan
Possamaï † May 27, 2016
Abstract
In a framework close to the one developed by Holmström and Milgrom [44], we studythe optimal contracting scheme between a Principal and several Agents. Each hired Agentis in charge of one project, and can make efforts towards managing his own project, as wellas impact (positively or negatively) the projects of the other Agents. Considering economicagents in competition with relative performance concerns, we derive the optimal contractsin both first best and moral hazard settings. The enhanced resolution methodology reliesheavily on the connection between Nash equilibria and multidimensional quadratic BSDEs.The optimal contracts are linear and each agent is paid a fixed proportion of the terminalvalue of all the projects of the firm. Besides, each Agent receives his reservation utility,and those with high competitive appetence are assigned less volatile projects, and shall evenreceive help from the other Agents. From the principal point of view, it is in the firm interestin our model to strongly diversify the competitive appetence of the Agents.
Key words:
Principal multi-agents problems, relative performance, Moral hazard, competition,Nash equilibrium, Multidimensional quadratic BSDEs.
AMS 2010 subject classification:
JEL classification:
C61, C73, D82, J33, M52
By and large, it has now become common knowledge among the economists, that almost ev-erything in economics was to a certain degree a matter of incentives: incentives to work hard,to produce, to study, to invest, to consume reasonably... Starting from the 70s, the theoryof contracts evolved from this acknowledgment and the fact that such situations could not bereproduced using the general equilibrium theory. In the corresponding typical situation, a Prin-cipal (who takes the initiative of the contract) is (potentially) imperfectly informed about theactions of an Agent (who accepts or rejects the contract). The goal is to design a contractthat maximizes the utility of the Principal while that of the Agent is held to a given level. Ofcourse, the form of the optimal contracts typically depends on whether these actions are observ-able/contractible or not, and on whether there are characteristics of the Agent that are unknown ∗ Université Paris-Est Marne-la-Vallée & Projet MathRisk INRIA, [email protected]. Research par-tially supported by the ANR grant LIQUIRISK and the Chair Finance and Sustainable Development. † CEREMADE, Université Paris Dauphine, [email protected].
1o the Principal. These problems are fundamentally linked to designing optimal incentives, andare therefore present in a very large number of situations (see Bolton and Dewatripont [4] orLaffont and Martimort [56] for many examples).The easiest problem corresponds to the case were the Principal is actually perfectly informedabout the actions of the Agent, and just has to find a way to optimally share the risks associatedto the project he has hired the Agent for, between the two of them: this is the so-called risk-sharing problem or first-best. Early studies of the risk-sharing problem can be found, amongothers, in Borch [5], Wilson [85] or Ross [73]. Since then, a large literature has emerged, solvingvery general risk-sharing problems, for instance in a framework with several Agents and recursiveutilities (see Duffie et al. [23]), or for studying optimal compensation of portfolio managers (seeOu-Yang [67] or Cadenillas et al. [8]).A more complicated situation arises in the so-called moral hazard case, or second-best, where thePrincipal cannot observe (or contract upon) the actions chosen by the Agent. For a long time,these problems were only considered in discrete-time or static settings , which are in generalquite hard to solve, and one had to wait for the seminal paper by Holmström and Milgrom [44]to witness the treatment of specific moral hazard problems in a continuous time framework.Their work was generalized by Schättler and Sung [77, 78], Sung [80, 81], Müller [62, 63], andHellwig and Schmidt [41], using a dynamic programming and martingales approach, which isclassical in stochastic control theory (see also the survey paper by Sung [82] for more references).This approach has then been extended to a very general framework in recent works by Cvitanić,Possamaï and Touzi [10, 11] . Yet another recent seminal paper is the one by Sannikov [75], whofinds a tractable model with a random time of retiring the Agent and with continuous payments,rather than a lump-sum payment at the terminal time. Since then, a growing literature extendingthe above models has emerged, see the illuminating survey paper [76] for a quite comprehensivelist of references.Another possible extension of the moral hazard problem lies in considering that the Principalno longer hires just one Agent, but several of them, to manage one or several projects onhis behalf, while being able to interact with each other. This is the so-called multi-Agentproblem. Early works in that direction, again in one-period frameworks, include Holmström[43], Mookherjee [61], Green and Stokey [37], Harris et al. [40], Nalebuff and Stiglitz [64] orDemski and Sappington [20]. As far as we know, the first extension to continuous time is dueto Koo et al. [52], who considered roughly the same model as Holmström [43], by using themartingale approach described above. Of course, as soon as one starts to consider contractingsituation involving several agents, the question of these agents comparing themselves to eachother becomes quite relevant. Hence, contemporary to the latter studies, several researcherstried to understand the impact of the so-called inequity aversion, as formulated by Fehr andSchmidt [31], on agency costs. The idea is that agents working in a firm dislike inequity, in thesense that an agent suffers a utility loss if another agent conducting a similar task receives a For very early moral hazard models, introducing the so-called first-order approach and then later its rigorousjustification, see Zeckhauser [87], Spence and Zeckhauser [79], or Mirrlees [58, 59, 60], as well as the seminalpapers by Grossman and Hart [38], Jewitt, [48], Holmström [42] or Rogerson [72]. Another, and in some cases more general, approach is to use the so-called stochastic maximum principle andFBSDEs to characterize the optimal compensation. This is the strategy used by Williams [84] and Cvitanić,Wan and Zhang [12, 13], and more recently, by Djehiche and Hegelsson [21, 22]. We also refer the reader to theexcellent monograph of Cvitanić and Zhang [14] for a systematic presentation of this approach.
Notations:
Let N ⋆ := N \ { } and let R ⋆ + be the set of real positive numbers. Throughout thispaper, for every p -dimensional vector b with p ∈ N ⋆ , we denote by b , . . . , b p its coordinates, forany ≤ i ≤ p , by b − i ∈ R p − the vector obtained by suppressing the i th coordinate of b , andfor p > b − i := 1 p − p X j =1 , j = i b j . For α, β ∈ R p we also denote by α · β the usual inner product, with associated norm k·k , whichwe simplify to | · | when p is equal to . We also let p be the vector of size p whose coordinatesare all equal to . For any ( l, c ) ∈ N ⋆ × N ⋆ , M l,c ( R ) will denote the space of l × c matriceswith real entries. Elements of the matrix M ∈ M l,c will be denoted by ( M i,j ) ≤ i ≤ l, ≤ j ≤ c , andthe transpose of M will be denoted by M ⊤ . We identify M l, with R l . When l = c , we let4 l ( R ) := M l,l ( R ) . For any x ∈ M l,c ( R ) , and for any ≤ i ≤ l and ≤ j ≤ c , x i, : ∈ M ,c ( R ) and x : ,j ∈ R l will denote respectively the i th row and the j th column of M . Moreover, for any x ∈ M l,c ( R ) and any ≤ j ≤ c , x : , − j ∈ M l,c − will denote the matrix x without the j th column.For any x ∈ R p , diag( x ) ∈ M p ( R ) will stand for the matrix whose diagonal is x and for whichoff-diagonal terms are , and I p will be the identity matrix in M p ( R ) . For any x ∈ M l,c ( R ) andany y ∈ R l , we also define, for any i = 1 , . . . , c , y ⊗ i x ∈ M l,c +1 ( R ) as the matrix whose column j = 1 , . . . , i − is equal to the j th column of x , whose column j = i + 1 , . . . , c + 1 is equal tothe ( j − th column of x , and whose i th column is y . We also abuse notations and denote forany x ∈ M p ( R ) by k x k the operator norm of x associated to the Euclidean norm on R p . Thetrace of a matrix M ∈ M l ( R ) will be denoted by Tr [ M ] .For any finite dimensional vector space E , with given norm k·k E , we also introduce the so-called Morse-Transue space on a given probability space (Ω , F , P ) (we refer the reader to themonographs [69, 70] for more details), defined by M φ ( E ) := { ξ := Ω −→ E, measurable , E [ φ ( aξ )] < + ∞ , for any a ≥ } , (1.1)where φ : E −→ R is the Young function, i.e. φ : x exp( k x k E ) − . Then, if M φ ( E ) isendowed with the norm k ξ k φ := inf { k > , E [ φ ( ξ/k )] ≤ } , is a (non-reflexive) Banach space. This opening section sets up properly the problem of interest. We first define the dynamics ofthe firm. Second, we model the impact of the economic choices made by the system of Agent.We then define properly the set of admissible strategies for the Agents in such system, and wefinally indicate the objective function of each Agent, as well as the one of the Principal.
We consider a model where a Principal wishes to hire N ≥ Agents, in order to take care of N different projects. Each Agent, if hired, will have the responsibility of a risky project. In orderto define precisely the outcome of the actions chosen by each Agent, let us start by fixing somenotations.We fix a deterministic time horizon T > . We work on a given probability space (Ω , F , P ) carrying an N -dimensional Brownian motion W . Each component of W will drive the noiseassociated to one project of the firm. We denote by F := ( F t ) ≤ t ≤ T the (completed) naturalfiltration of W . As is customary in the Principal/Agent literature, we will work under theso-called weak formulation of the problem.To define this rigorously, let us start by defining the so-called output process X of the firm,which is R N -valued, X t := Z t Σ s dW s , ≤ t ≤ T, a.s., (2.1)where for any t ∈ [0 , T ] , Σ t ∈ M N ( R ) . Each component of the vector X corresponds to oneproject of the firm and the matrix Σ characterizes their correlation. Our assumption on Σ is asfollows 5 ssumption 2.1. The map
Σ : [0 , T ] −→ M N ( R ) is ( Borel ) measurable, bounded and such thatfor any t ∈ [0 , T ] , Σ t is invertible. We will consider that each Agent will be assigned a project but can impact both his own projectas well as the projects of the other Agents. Hence, the controls of all the Agents will be givenby a matrix a ∈ M N ( R ) , such that for any ≤ i, j ≤ N , a i,j represents the action of Agent j for the project managed by Agent i . In other words, each Agent can choose to make effortstowards managing his own project, but he can also decide to impact (positively or negatively)the projects of the other Agents.We also introduce for any ≤ i ≤ N the maps b i : [0 , T ] × R N × R N −→ R , which are assumedto satisfy the following Assumption 2.2.
For every i = 1 , . . . , N , and every ( t, x ) ∈ [0 , T ] × R N , the maps a b i ( t, a, x ) are C , and for every a ∈ R N , the maps x b i ( t, a, x ) are uniformly Lipschitzcontinuous. We also assume that for some some constant C > , (cid:12)(cid:12) b i ( t, a, x ) (cid:12)(cid:12) ≤ C (1+ k a k + k x k ) , (cid:13)(cid:13) ∇ a b i ( t, a, x ) (cid:13)(cid:13) ≤ C, for any ( t, a, x ) ∈ [0 , T ] × R N × R N . (2.2)For any ( t, a, x ) ∈ [0 , T ] × M N ( R ) × R N , we also denote by b ( t, a, x ) the vector of R N whose i thcoordinate is b i ( t, a : ,i , x ) .Notice that by Assumption 2.2, for any M N ( R ) -valued and F -progressively measurable process a , the following SDE Z t = Z t b ( s, a s , Z s ) ds + Z t Σ s dW s , t ∈ [0 , T ] , P − a.s., (2.3)admits a unique strong solution denoted X a .For any M N ( R ) -valued F -progressively measurable processes a satisfying E (cid:18)Z T b ( s, a s , X s ) · Σ − s dW s (cid:19) has moments of order (1 + ǫ ) for some ǫ > , (2.4)we can then define a new probability measure P a on (Ω , F ) , equivalent to P , with d P a d P := E (cid:18)Z T b ( s, a s , X s ) · Σ − s dW s (cid:19) . By Girsanov’s theorem, we then know that for any a ∈ A , the following R N -valued process W at := W t − Z t Σ − s b ( s, a s , X s ) ds, ≤ t ≤ T, a.s., is a Brownian motion under P a , so much so that we can rewrite (2.1), for any a ∈ A , as X t = Z t b ( s, a s , X s ) ds + Z t Σ s dW as , ≤ t ≤ T, a.s.
Moreover, it is clear that P a coincides with the probability measure e P a := P ◦ ( X a ( X a · ) ) − obtainedfrom the strong solution of (2.3). 6 .3 The admissible efforts for the Agents We introduce for any ≤ i ≤ N , the cost function of the Agent i which we denote k i :[0 , T ] × R N × R N −→ R + . We note for any ( t, a, x ) ∈ [0 , T ] × M N ( R ) × R N , by k ( t, a, x ) ∈ R N the vector whose i th coordinate is k i ( t, a : ,i , x ) . Our standing assumption on the vector costfunction k is the following. Assumption 2.3.
For any ( t, x ) ∈ [0 , T ] × R N , the map a k ( t, a, x ) is C , and has each ofits coordinates which are increasing and strictly convex. Moreover, we have for some constants C > and ℓ ≥ k a k→ + ∞ k k ( t, a, x ) kk a k = + ∞ , k k ( t, a, x ) k ≤ C (cid:16) k a k ℓ + k x k (cid:17) , k∇ a k ( t, a, x ) k ≤ C (cid:16) k a k ℓ − (cid:17) . We are now in position to define the set of admissible strategies A for the system of Agents. Inour framework of study, each Agent i is restricted to choose a control in a given subset A i of R N . The set A of admissible controls is then defined as the set of F -adapted processes a , whichare M N ( R ) -valued, such that for any ≤ i ≤ N , ( a i, : ) ⊤ takes values in A i , (2.4) holds and R T b ( s, a s , X s ) ds as well as R T k ( s, a s , X s ) ds are valued in M φ ( R N ) . Recall that M φ ( R N ) is theMorse-Transue space associated to R N , see (1.1). Now that the set of admissible strategies of the agents has been established, let us turn to thedesign of their objective function. We assume that the Agents derive utility from two sources.First, from the salary they receive from the contract, diminished by the cost induced by theirworking effort. Second, the Agents derived utility from the success of their project in comparisonto the other ones. In our model, the main motivation for the interaction between the Agentscomes from this feature, which makes them compare to each other.More precisely, we assume that the utilities of the Agents are exponential and that, for any ≤ i ≤ N , given N contracts ξ := ( ξ , . . . , ξ N ) ⊤ and a choice of actions a ∈ A made by all theAgents, the utility of Agent i is U i ( a : ,i , a : , − i , ξ i ) := E P a (cid:20) U Ai (cid:18) ξ i + Γ i ( X T ) − Z T k i ( s, a : ,is , X s ) ds (cid:19)(cid:21) , (2.5)with U Ai ( x ) := − exp (cid:0) − R iA x (cid:1) , x ∈ R , Γ i : R N −→ R , where R Ai > represents the risk-aversion of Agent i , and the map Γ i corresponds to Agent i comparing his performance with the performances of the other Agents. This map Γ i can bequite general and a typical example would be Γ i ( x ) := γ i (cid:0) x i − ¯ x − i (cid:1) , x ∈ R N , (2.6)where γ i is a given non-negative constant, so called competition index of Agent i . This settingcorresponds to the case where each Agent compares his performance to the average of the otherAgents performances. The higher γ i is, the more competitive Agent i will be.7n general, we assume that the comparison map Γ satisfies Assumption 2.4.
For any ≤ i ≤ N , the maps Γ i = R N −→ R are ( Borel ) measurable andsatisfy, for some C > | Γ i ( x ) | ≤ C (1 + k x k ) , x ∈ R N . Now that the incentives of the Agents are well understood, we can turn to the design of thoseof the Principal. In our setting, the Principal offers simultaneously a contract to each Agent attime , and he can commit to any such contract. For any ≤ i ≤ N , a contract for Agent i will then be represented as a real valued random variable ξ i , which, for now, we only assume tobe F T -measurable, representing the amount of money that Agent i will receive at time T at theend of the contract. Observe that there are no intertemporal payments.Given such vector of terminal payment ξ , the system of Agents in interaction will choose someresponse actions a ∈ A . Hence, each agent will obtain from the game at time the utility value U i ( a, ξ i ) . In order to ensure that each Agent agrees to enter the game, the Principal will restricthis contracts offers to those such that each agent i receives at least his reservation utility denotedU i , i.e such that U i ( a, ξ i ) ≥ U i , ≤ i ≤ N. (2.7)Besides, for given contracts ( ξ i ) ≤ i ≤ N and a given choice of actions a ∈ A made by the Agents,the Principal derives utility from the terminal values of the projects as follows U P ( a, ξ ) := E P a [ − exp ( − R P ( X T − ξ ) · N )] , (2.8)where R P > is the risk-aversion of the Principal.Of course, for all of this to be meaningful, we still need to impose conditions on a and ξ so that(2.5) and (2.8) are well defined. We will take care of this problem later on when defining the setof admissible contracts.To sum up, the Principal will choose N contracts ( ξ i ) ≤ i ≤ N , leading to a response effort a of thesystem of Agents, such that each Agent wishes to enter the contract, i.e. (2.7) is satisfied, andwhich should be optimal according to the enhanced criterion (2.8). Depending on how muchinformation is available for the Principal, we now split our study into the so-called correspondingfirst-best and second-best settings. In this section, we concentrate our attention to the so-called first-best framework, where there isno moral hazard and the Principal can actually choose directly both the contracts ( ξ i ) ≤ i ≤ N aswell as the actions of the Agents. As far as we know, this problem in our framework has neverbeen solved in the literature.We first rewrite the problem of the Principal in a more tractable stochastic control form. Then,we provide in Section 3.2 a representation of the optimal contract in full generality. We finallyfocus more closely in Section 3.3 on the more tractable setting with linear comparison map Γ ofthe form (2.6). 8 .1 Stochastic control reformulation of the Principal problem In this case, the set of admissible contracts C F B will be defined as C F B := n ξ, F T -measurable with ξ ∈ M φ ( R N ) o . It is then clear by construction that for any ( a, ξ ) ∈ A × C F B , the quantities U i ( a : ,i , a : , − i , ξ i ) and U P ( a, ξ ) are well-defined.Given some reservation utility levels ( U i ) ≤ i ≤ N (which are all negative) chosen by the Agents,the problem of the Principal is then U P,F B := sup a ∈A sup ξ ∈C F B ( U P ( a, ξ ) + N X i =1 ρ i U i ( a : ,i , a : , − i , ξ i ) ) , (3.1)where the ρ i > are the Lagrange multiplier associated to the participation constraints.Let us start with the maximization with respect to ξ . In order to do so, for any a ∈ A , let usconsider the following map Ξ a : M φ ( R N ) −→ R defined by Ξ a ( ξ ) := E P a " − e − R P ( X T − ξ ) · N − N X i =1 ρ i e − R iA ( ξ i +Γ i ( X T ) − R T k i ( s,a : ,is ,X s ) ds ) . Since ρ i > for any ≤ i ≤ N , it can easily be seen that Ξ a is continuous, strictly convex,proper, and Gâteaux differentiable, with Gâteaux derivative given, for any h ∈ M φ ( R N ) , by D Ξ a ( ξ )[ h ] = E P a h − R P h · N e − R P ( X T − ξ ) · N + N X i =1 ρ i R iA h i e − R iA ( ξ i +Γ i ( X T ) − R T k i ( s,a : ,is ,X s ) ds ) . For any a ∈ A , let introduce ξ ⋆ ( a ) by ( ξ ⋆ ( a )) i := 1 R iA log (cid:18) ρ i R iA R P (cid:19) − Γ i ( X T ) + Z T k i ( s, a : ,is , X s ) ds + R P R iA ( X T · N − ξ ⋆ ( a ) · N ) , together with ξ ⋆ ( a ) · N := R A R A + N R P (cid:18) R P NR A X T − Γ( X T ) + Z T k ( s, a s , X s ) ds (cid:19) · N + R A R A + N R
P N X i =1 R iA log (cid:18) ρ i R iA R P (cid:19) , (3.2)where Γ( X T ) ∈ R N is the vector whose i th coordinate is Γ i ( X T ) , and where R A := N P Ni =1 1 R iA . Then, for any h ∈ M φ ( R N ) , we have D Ξ a ( ξ ⋆ ( a ))[ h ] = 0 , so that this ξ ⋆ ( a ) attains the minimumof Ξ a and is therefore optimal. 9lugging these expressions back into the Principal problem, we obtain that the principal problemrewrites in a stochastic control form as R A + N R P R A N Y i =1 (cid:18) ρ i R iA R P (cid:19) RP RARiA ( RA + NRP ) sup a ∈A E P a " − e RP RARA + NRP ( R T k ( s,a s ,X s ) ds − X T − Γ( X T ) ) · N . We will first consider the case of a general interaction function Γ and then tackle a benchmarkcase where Γ is linear, for which the solution is much easier to find. Γ Under the form above, the problem of the Principal is now a classical stochastic control problem(under weak formulation), with controlled state process X , whose drift only is controlled. Denoteby A := Q Ni =1 A i . It is then a classical result (see for instance [25, 26], or similar comparisonarguments in Section 4.1.2 below) that we have U P,F B = − R A + N R P R A N Y i =1 (cid:18) ρ i R iA R P (cid:19) RP RARiA ( RA + NRP ) exp (cid:18) − R P R A R A + N R P Y (cid:19) , where ( Y, Z ) denotes the maximal solution of the following BSDE Y t = ( X T + Γ( X T )) · N + Z Tt F ( s, X s , Z s ) ds − Z Tt Z s · Σ s dW s , t ∈ [0 , T ] , P − a.s., (3.3)where the generator F is given by F ( t, x, z ) := sup a ∈ A { b ( t, a, x ) · z − k ( t, a, x ) · N } − R P R A R A + N R P ) k Σ t z k . We still need to justify why the BSDE (3.3) indeed admits a maximal solution. First, byAssumptions 2.1 and 2.4, we know that the terminal condition has linear growth w.r.t. X T and thus admits exponential moments of any order under P (remember that, under P , X T isGaussian). Moreover, by Lemma 4.1 below, we have that (cid:12)(cid:12)(cid:12)(cid:12) sup a ∈ A { b ( t, a, x ) · z − k ( t, a, x ) · N } (cid:12)(cid:12)(cid:12)(cid:12) ≤ C (cid:16) k z k ℓℓ − (cid:17) . Hence, since ℓ ≥ , F has at most quadratic growth in z . Since F ( t, x,
0) = 0 , the existence ofa solution is given by [6]. However, sufficient (but not necessary ) condition for the existenceof a maximal solution is that F is bounded from above by a map which has linear growth in z (see [1]), or that F is concave or convex in z (see [7]). Such a condition requires the followingadditional assumption Another condition would be that F is actually purely quadratic in z , that is to say F ( t, x, z ) = f ( t, x ) + k γ ( t ) z k , for some maps f : [0 , T ] × R N −→ R and γ : [0 , T ] −→ M N ( R ) , in which case one can easily show, using anexponential transformation, that the BSDE actually has a unique solution. This would be the case if for instance b was chosen linear in a and k appropriately quadratic in a (see the next section). ssumption 3.1. Either the map z F ( t, x, z ) is convex or concave for any ( t, x ) ∈ [0 , T ] × R N , or there exists some C > such that for any ( t, x, z ) ∈ [0 , T ] × R N × R N F ( t, x, z ) ≤ C (1 + k x k + k z k ) . Remark 3.1.
Whenever the set A is compact, then Assumption 3.1 is automatically satisfied. Denote, for any ( t, z, x ) ∈ [0 , T ] × R N × R N , by ˜ a ⋆ ( t, x, z ) ∈ M N one of the maximizers of themap a b ( t, a, x ) · z − N · k ( t, a, x ) . Notice that such a maximizer is well defined since byAssumptions 2.2 and 2.3, k has superlinear growth at infinity, while b has linear growth (see(2.2)), so that the map considered here is coercive. By a classical measurable selection argument,we deduce that the corresponding predictable process a ⋆t := ˜ a ⋆ ( t, X t , Z t ) , defined dt × d P − a.e. , isthe optimal effort for the Agents, chosen by the Principal, and that the corresponding contract ξ ⋆ ( a ⋆ ) is optimal, provided they are both admissible. So as not to complicate further ourpresentation, we refrain from giving general conditions under which this holds true.Using this contract and this action, one still needs to choose the Lagrange multipliers ρ i suchthat U i (( a ⋆ ) : ,i , ( a ⋆ ) : , − i , ( ξ ⋆ ( a ⋆ )) i ) = U i . (3.4)We have thus obtained the following verification type result Theorem 3.1.
Let Assumptions 2.1, 2.2, 2.3, 2.4 and 3.1 hold and assume furthermore that a ⋆ ∈ A and ξ ⋆ ( a ⋆ ) ∈ C F B . Then, the contract ξ ⋆ ( a ⋆ ) chosen so that (3.4) holds is optimal forthe Principal, with a recommended level of effort a ⋆ . Instead of elaborating further on technical conditions ensuring the admissibility of the effort andcontract, we choose to specialize to a linear-quadratic framework with simpler dynamics, forwhich the BSDE (3.3) can be solved explicitly.
We now consider a simplified linear-quadratic setting, i.e. where the drift of the output processand the cost function are respectively linear and quadratic with respect to the control a . Namely,we work under the additional assumption: Assumption 3.2.
For any , . . . , N , we have A i = R N , the maps b i and k i have the form b i ( t, a, x ) = B N · a + ˜ b i ( t, x ) , k i ( t, a, x ) = K k a k + ˜ k i ( t, x ) , ∀ ( t, a, x ) ∈ [0 , T ] × R N × R N , for some B ∈ R , K ∈ (0 , + ∞ ) , and some maps ˜ b i : [0 , T ] × R N −→ R and ˜ k i : [0 , T ] × R N −→ R + .Moreover, we assume that the volatility matrix Σ t does not depend on time and is a multiple ofthe identity matrix, that is Σ t = σI N , for some σ ∈ (0 , + ∞ ) . Remark 3.2.
Observe in particular that Assumption 3.1 is a direct consequence of Assumption3.2. Similarly the linear upper bound on the driver F required in Assumption 3.1 is automaticallyvalid if Assumptions 2.2, 2.3, and 3.2 are satisfied.
11s usual, we denote by ˜ b ( t, x ) (resp. ˜ k ( t, x ) ) the vector of R N whose i th component is ˜ b i ( t, x ) (resp. ˜ k i ( t, x ) ). Under Assumption 3.2, direct computations show that F ( t, x, z ) = N | B | k − | σ | R P R A R A + N R P ! N X i =1 (cid:12)(cid:12) z i (cid:12)(cid:12) b ( t, x ) z − ˜ k ( t, x ) , which corresponds to an optimal effort a ⋆ ( z ) ∈ M N ( R ) such that a ⋆ ( z ) := Bk z ⊤ N , z ∈ R N . Recall that ( Y, Z ) is the maximal solution to the BSDE (3.3) and define η := N | B | k | σ | − R P R A R A + N R P ! , P t := exp ( ηY t ) , Q t := σηP t Z t , t ∈ [0 , T ] . A simple application of Itô’s formula together with (3.3) gives P t = e η ( X T +Γ( X T )) · N + Z Tt ˜ b ( s, X s ) σ · Q s − ηP s ˜ k ( s, X s ) · N ! ds − Z Tt Q s · dW s , that is ( P, Q ) solves a simple linear BSDE. In particular, defining a new probability measure e P by d e P d P := E Z T ˜ b ( s, X s ) σ · dW s ! , we rewrite P t = E e P (cid:20) exp (cid:18) η N · (cid:18) X T + Γ( X T ) − Z Tt ˜ k ( s, X s ) ds (cid:19)(cid:19)(cid:12)(cid:12)(cid:12)(cid:12) F t (cid:21) , so that Y t = 1 η log (cid:18) E e P (cid:20) exp (cid:18) η N · (cid:18) X T + Γ( X T ) − Z Tt ˜ k ( s, X s ) ds (cid:19)(cid:19)(cid:12)(cid:12)(cid:12)(cid:12) F t (cid:21)(cid:19) . In order to have access to the optimal effort a ⋆ , we need to reinforce Assumption 2.4. Assumption 3.3.
The Borel measurable map Γ is Lipschitz-continuous. Then, it is a well known result (see for instance Proposition 5.3 in [26]), that a version of Q t isgiven by D t P t , where D is the Malliavin differentiation operator. Since P is given as a conditionalexpectation of the composition of a smooth and a Lipschitz-continuous function, we computedirectly using the chain rule of Malliavin calculus that Q t = E e P (cid:20) η (cid:18) D t X T + Γ ′ ( X T ) · N D t X T − Z Tt ˜ k x ( s, X s ) · N D t X s ds (cid:19) × exp (cid:18) η N · (cid:18) X T + Γ( X T ) − Z Tt ˜ k ( s, X s ) ds (cid:19)(cid:19)(cid:12)(cid:12)(cid:12)(cid:12) F t (cid:21) + E e P (cid:20) (cid:18) σ Z Tt ˜ b x ( s, X s ) D t X s dW s − σ Z Tt ˜ b x ( s, X s )˜ b ( s, X s ) D t X s (cid:19) × exp (cid:18) η N · (cid:18) X T + Γ( X T ) − Z Tt ˜ k ( s, X s ) ds (cid:19)(cid:19)(cid:12)(cid:12)(cid:12)(cid:12) F t (cid:21) , t ∈ [0 , T ] and Lebesgue almost every x ∈ R N , Γ ′ ( x ) denotes the vector of R N whose i th component is (Γ i ) ′ ( x ) and ˜ k x ( t, x ) denotes the vector of R N whose i th component is ˜ k ix ( t, x ) . We deduce that a ⋆ ( Z t ) = Bkη (cid:18) E e P (cid:20) η (cid:18) D t X T + Γ ′ ( X T ) · N D t X T − Z Tt ˜ k x ( s, X s ) · N D t X s ds (cid:19) × exp (cid:18) η N · (cid:18) X T + Γ( X T ) − Z Tt ˜ k ( s, X s ) ds (cid:19)(cid:19)(cid:12)(cid:12)(cid:12)(cid:12) F t (cid:21) + E e P (cid:20) (cid:18) σ Z Tt ˜ b x ( s, X s ) D t X s dW s − σ Z Tt ˜ b x ( s, X s )˜ b ( s, X s ) D t X s (cid:19) × exp (cid:18) η N · (cid:18) X T + Γ( X T ) − Z Tt ˜ k ( s, X s ) ds (cid:19)(cid:19)(cid:12)(cid:12)(cid:12)(cid:12) F t (cid:21)(cid:19) × (cid:18) E e P (cid:20) exp (cid:18) η N · (cid:18) X T + Γ( X T ) − Z Tt ˜ k ( s, X s ) ds (cid:19)(cid:19)(cid:12)(cid:12)(cid:12)(cid:12) F t (cid:21)(cid:19) − ⊤ N . Finally, if ˜ k and ˜ b do not depend on x , we can further simplify the above expression, using thefact that under e P , X T is, conditionally on F t , an N -dimensional Gaussian random variable withmean X t + R Tt ˜ b ( s ) ds and variance-covariance matrix σ ( T − t ) I N Y t = X t · N − Z Tt ˜ k ( s ) · N ds − N η log (cid:0) πσ ( T − t ) (cid:1) + 1 η log Z R N e η N · ( x +Γ( x + X t )) − k x − R Tt ˜ b ( s ) ds k σ T − t ) dx . Similarly, we have Q t = ησ N Z R N (cid:16) ′ ( x + X t ) · N (cid:17) e η N · x + X t +Γ( x + X t ) − R Tt ˜ k ( s ) ds − k x − R Tt ˜ b ( s ) ds k σ T − t ) ! dx, so that a ⋆ ( Z t ) = Bσk Z R N (cid:16) ′ ( x + X t ) · N (cid:17) e η N · x + X t +Γ( x + X t ) − R Tt ˜ k ( s ) ds − k x − R Tt ˜ b ( s ) ds k σ T − t ) ! dx Z R N e η N · x + X t +Γ( x + X t ) − R Tt ˜ k ( s ) ds − k x − R Tt ˜ b ( s ) ds k σ T − t ) ! dx . We summarize all the above in the following theorem.
Theorem 3.2.
Let Assumptions 2.2, 2.3, 3.2 and 3.3 hold. Then, if the contract ξ ⋆ ( a ∗ ) ( wherethe ρ i are chosen so that (3.4) holds ) belongs to C F B , this contract is optimal for the Principal, here the optimal effort a ⋆ is given by the following process a ⋆t := Bkη (cid:18) E e P (cid:20) η (cid:18) D t X T + Γ ′ ( X T ) · N D t X T − Z Tt ˜ k x ( s, X s ) · N D t X s ds (cid:19) × exp (cid:18) η N · (cid:18) X T + Γ( X T ) − Z Tt ˜ k ( s, X s ) ds (cid:19)(cid:19)(cid:12)(cid:12)(cid:12)(cid:12) F t (cid:21) + E e P (cid:20) (cid:18) σ Z Tt ˜ b x ( s, X s ) D t X s dW s − σ Z Tt ˜ b x ( s, X s )˜ b ( s, X s ) D t X s (cid:19) × exp (cid:18) η N · (cid:18) X T + Γ( X T ) − Z Tt ˜ k ( s, X s ) ds (cid:19)(cid:19)(cid:12)(cid:12)(cid:12)(cid:12) F t (cid:21)(cid:19) × (cid:18) E e P (cid:20) exp (cid:18) η N · (cid:18) X T + Γ( X T ) − Z Tt ˜ k ( s, X s ) ds (cid:19)(cid:19)(cid:12)(cid:12)(cid:12)(cid:12) F t (cid:21)(cid:19) − ⊤ N . which is simplified, when ˜ k and ˜ b do not depend on x to a ⋆t = Bσk Z R N (cid:16) ′ ( x + X t ) · N (cid:17) e η N · x + X t +Γ( x + X t ) − R Tt ˜ k ( s ) ds − k x − R Tt ˜ b ( s ) ds k σ T − t ) ! dx Z R N e η N · x + X t +Γ( x + X t ) − R Tt ˜ k ( s ) ds − k x − R Tt ˜ b ( s ) ds k σ T − t ) ! dx . It is worth noticing that our framework allow for the consideration of rather general Γ functions,allowing for example to consider Agents interested in their (smoothed approximate) rankingwithin the population of Agents. Nevertheless, in order to obtain more explicit representation ofthe solution, we now focus on a particular form of comparison criterium in between the Agents,relying on a linear Γ function. Γ In this section, we do not focus anymore on a linear-quadratic setting but specialize the discussionto another solvable framework by imposing:
Assumption 3.4.
We have Γ i ( x ) = x i − ¯ x − i , b i ( t, a, x ) = b i ( t, a ) , k i ( t, a, x ) = k i ( t, a ) , ≤ i ≤ N, ( a, x ) ∈ R N × R N . In this case, we have Γ( X T ) · N = (cid:0) γ − γ − (cid:1) · X T , where γ ∈ R N is the vector whose i th coordinate is γ i and where γ − is such that for any ≤ i ≤ N , ( γ − ) i := γ − i . Denote p := γ − − N − γ . The Principal problem can then be rewritten R A + N R P R A e (cid:18) RP RARA + NRP (cid:19) R T k Σ t p k dt N Y i =1 (cid:18) ρ i R iA R P (cid:19) RP RARiA ( RA + NRP ) × sup a ∈A E P a " −E (cid:18) R P R A R A + N R P p · Z T Σ t dW at (cid:19) e RP RARA + NRP R T ( p · b ( s,a s )+ N · k ( s,a s )) ds . a ⋆ ( t ) be any (deterministic) minimizer of the map a p · b ( t, a ) + N · k ( t, a ) . Since W a is a Brownian motion under P a , it is clear that the stochastic exponential which appears aboveis a uniformly integrable martingale. Hence, we deduce that U P,F B ≤ R A + N R P R A e (cid:18) RP RARA + NRP (cid:19) R T k Σ t p k dt N Y i =1 (cid:18) ρ i R iA R P (cid:19) RP RARiA ( RA + NRP ) × exp (cid:18) R P R A R A + N R P Z T ( p · b ( s, a ⋆s ) + N · k ( s, a ⋆s )) ds (cid:19) . But this upper bound can easily be attained by choosing action a ⋆ and contract ξ ⋆ ( a ⋆ ) as in (3.2).Moreover, since a ⋆ is deterministic, it obviously belongs to A , and we also have ξ ⋆ ( a ⋆ ) ∈ C F B ,since it is linear in X which has exponential moments of any order.Finally, let us compute the utility that Agent i obtains from this contract. Recalling (2.5), weget U i (( a ⋆ ) : ,i , ( a ⋆ ) : , − i , ( ξ ⋆ ( a ⋆ )) i )= R P ρ i R iA N Y j =1 ρ j R jA R p ! RP RARjA ( RA + NRP ) e RP RARA + NRP N · R T k ( s,a ⋆s ) ds × E P a⋆ (cid:20) − exp (cid:18) R P R A R A + N R P p · X T (cid:19)(cid:21) = − R P ρ i R iA N Y j =1 ρ j R jA R p ! RP RARjA ( RA + NRP ) e RP RARA + NRP N · R T k ( s,a ⋆s ) ds + RP RARA + NRP R T p · b ( s,a ⋆s ) ds × e (cid:18) RP RARA + NRP (cid:19) R T k Σ t p k dt . We therefore need to determine the Lagrange multipliers ( ρ i ) ≤ i ≤ N so that we have Y ≤ j ≤ N, j = i ρ RP RARjA ( RA + NRP ) j ρ RP RARiA ( RA + NRP ) − i = − R iA R P A U i , ≤ i ≤ N, (3.5)where A > is defined by A := e − RP RARA + NRP N · R T k ( s,a ⋆s ) ds − RP RARA + NRP R T p · b ( s,a ⋆s ) ds − (cid:18) RP RARA + NRP (cid:19) R T k Σ t p k dt . Then, if we define vectors ( B, R, log( ρ )) ∈ R N × R N × R N with B i := R A + N R P R P R A log (cid:18) − R iA R P A U i (cid:19) , R i := 1 R iA , log( ρ ) i := log( ρ i ) , then by taking logarithms on both sides of (3.5), we obtain that (3.5) is actually equivalent tosolving the linear system (cid:18) N R ⊤ − R A + N R P R P R A I N (cid:19) log( ρ ) = B.
15t can then be checked directly that (cid:18) N R ⊤ − R A + N R P R P R A I N (cid:19) − = − R P R A R A + N R P (cid:16) R P N R ⊤ + I N (cid:17) . Therefore, we finally have ρ i = − R P AR iA U i N Y j =1 " − R P AR jA U j RPRjA , ≤ i ≤ N. We have just proved the following
Theorem 3.3.
Let Assumptions 2.1, 2.2, 2.3 and 3.4 hold. Then an optimal contract ξ F B ∈ C
F B in the problem (3.1) , with reservation utilities ( U i ) ≤ i ≤ N ∈ ( −∞ , N , is given, for i = 1 , . . . , N ,by ξ iF B := R P R A R iA ( R A + N R P ) ( N + γ − γ − ) · X T − γ i (cid:16) X iT − X − iT (cid:17) + Z T k i ( s, ( a ⋆ ) : ,is ) ds + R P R A R iA ( R A + N R P ) Z T b ( s, a ⋆s ) · ( γ − − γ − N ) ds − R iA log( − U i )+ 12 R iA (cid:18) R P R A R A + N R P (cid:19) Z T (cid:13)(cid:13) Σ s ( γ − − γ − N ) (cid:13)(cid:13) ds. (3.6) where for any t ∈ [0 , T ] , the optimal action a ⋆t ∈ M N ( R ) is any minimizer of the map a ( γ − − γ − N ) · b ( t, a ) + N · k ( t, a ) . Moreover, the value function of the Principal is then U P,F B = − N Y i =1 "(cid:0) − U i (cid:1) − RPRiA e R P RA RA + NRP ) R T k Σ s ( γ − − γ − N ) k ds + R P R T ( k · N − b · ( N + γ − γ − )( s,a ⋆s ) ) ds . We specialize here the discussion to a setting where everything can be computed explicitly, inparticular the optimal actions of the Agents. For simplicity, we choose N = 2 , A = A = R ,as well as b ( t, a ) := a , − a , a , − a , ! , for any a := a , a , a , a , ! ∈ M ( R ) , and for some constants ( k , , k , , k , , k , ) ∈ ( R ⋆ + ) k ( t, a ) := k , (cid:12)(cid:12) a , (cid:12)(cid:12) + k , (cid:12)(cid:12) a , (cid:12)(cid:12) k , (cid:12)(cid:12) a , (cid:12)(cid:12) + k , (cid:12)(cid:12) a , (cid:12)(cid:12) ! , for any a := a , a , a , a , ! ∈ M ( R ) . In this setting, the vector p is simply given by p = γ − γ − γ − γ − ! . ptimal effort of the agents In our context, the strictly convex map that we need to minimize is f ( a ) := k , (cid:12)(cid:12) a , (cid:12)(cid:12) + k , (cid:12)(cid:12) a , (cid:12)(cid:12) + k , (cid:12)(cid:12) a , (cid:12)(cid:12) + k , (cid:12)(cid:12) a , (cid:12)(cid:12) − (1 + γ − γ )( a , − a , ) − (1 + γ − γ )( a , − a , ) . We have ∂f∂a , = k , a , − − γ + γ , ∂f∂a , = k , a , + 1 + γ − γ ,∂f∂a , = k , a , + 1 + γ − γ , ∂f∂a , = k , a , − − γ + γ , so that the optimal actions of the two Agents are a ⋆ := γ − γ k , γ − γ k , − γ − γ k , − γ − γ k , ! . Hence, if for instance Agent is much more competitive than Agent , so that γ > γ , thenAgent will work towards his project and will also work to decrease the value of the project ofAgent , while Agent will work to decrease the value of his own project and to increase thevalue of the project of Agent . Optimal recruitment scheme for the principal
Let us now ask ourselves the question of the optimal type of Agents that the Principal shouldhire. More precisely, given the choice between many Agents, what are the optimal parameters ( γ , γ ) for the Principal?From our general result, the problem for the Principal is then to maximize his value function,which is then equivalent to minimizing the map g ( γ , γ ) := (1 + γ − γ ) α + (1 + γ − γ ) α , where α := R P R A R A + 2 R P σ − (cid:18) k , + 1 k , (cid:19) , α := R P R A R A + 2 R P σ − (cid:18) k , + 1 k , (cid:19) . Then, it is clear that if α + α ≤ , the Principal would like to hire Agents with | γ − γ | −→ + ∞ , whereas if α + α > , the Principal wants to hire Agents with γ − γ = α − α α + α . Notice that this situation is optimal for the Principal, but also for the hired Agents, since inany case they receive their reservation utility. More importantly, let us emphasize that in ourmodel, the principal should hire agents with different competitively appetence profile. A firmhas economic gain from hiring Agents with diverse competitive profiles.17 .3.3 Economic interpretation
The general from of the optimal contract that we have obtained is given by ( ξ ⋆ ) i ( a ⋆ ) = C i + R P R A R iA ( R A + N R P ) ( N + γ − γ − ) · X T − γ i (cid:16) X iT − X − iT (cid:17) , for some constant C i , allowing to satisfy the participation constraint.Hence, the Principal penalizes each Agent with the amount − γ i ( X iT − X − iT ) , so as to suppressthe appetence for competition of the Agents. More precisely, Agents who do better than theaverage are penalized, while Agents who do worse are gratified, with the exact amount whichmakes them indifferent towards the performances of the other Agents. Competitive Agents arepaid less but each other Agent has incentives to work on the project of a competitive Agent.Moreover, each Agent is paid a positive part of each projects, the percentage depending on therisk aversion of the Agent, and of the universal vector R P R A R A + N R P ( N + γ − γ − ) . Hence, each Agent receives, for any ≤ i ≤ N , a fraction of the terminal value of the i th project,which is proportional to γ i − γ − i . This therefore means that if an Agent is particularlycompetitive, compared to the others, then all the Agents will receive a large part of his project,which gives them incentives not to work against the interest of this particular Agent. Conversely,if Agent i is not very competitive, then the other Agents could be penalized (if γ i − ( γ − ) i < ) by the terminal value of his project, which gives them incentives to reduce the value of hisproject as much as possible.Observe that the objective function of a competitive Agent is such that, whenever his projectsucceeds (in comparison to the others), he requires less salary for a similar utility value of thegame at time . Hence, we observe that a Principal should allocate competitive Agents toprojects with the highest probability of success, i.e. those with smallest volatility and thosewhich may benefit from the help of the other Agents. Similarly, it is worth noticing that it is inthe firm (i.e. the Principal) interest to hire Agents with diverse competition appetence profiles. In this section, we consider the so-called second-best problem when moral hazard exists. In sucha case, the Principal cannot observe the actions chosen by the Agents, and can only control thesalaries that he offers. The main difficulty here compared to the one-Agent case of Holmströmand Milgrom [44], is that, given a contract ξ , we have to be able to find an equilibrium resultingof the interactions between the Agents. Since the agents are playing simultaneously, we arelooking for a Nash equilibrium. The notion of equilibrium of interest is that of a Nash equilibrium. This calls for a first definition.For any i = 1 , . . . , N , and for any action a − i valued in M N,N − chosen by all the Agents j = i ,18e define the set A i ( a − i ) := (cid:8) ( a s ) ≤ s ≤ T , R N -valued, such that a ⊗ i a − i ∈ A (cid:9) . Definition 4.1.
Given a contract ξ , a Nash equilibrium for the N Agents is an action a ⋆ ( ξ ) ∈ A such that for any i = 1 , . . . , N , we have sup a ∈A i (( a ⋆ ) : − i ( ξ )) U i ( a, ( a ⋆ ) : , − i ( ξ ) , ξ i ) = U i (( a ⋆ ) : ,i ( ξ ) , ( a ⋆ ) : , − i ( ξ ) , ξ i ) . Since the uniqueness of a Nash equilibrium is more the exception than the rule, we also need toassume that the community of Agents has agreed on a common rule to choose between differentequilibria. More precisely, we are given a total order on R N , which we denote by (cid:23) . For, instancewe could consider the order defined by, for any ( x, y ) ∈ R N × R N x (cid:23) y, if N X i =1 U i ( x i ) ≤ N X i =1 U i ( y i ) , for some given utility function ( U i ) ≤ i ≤ N , which means that the community of Agents prefersthe Nash equilibria which maximize the total utility of the Agents. Moreover, we assume thatif the community of Agents is indifferent (for the order (cid:23) ) between several equilibriums, then italways chooses the ones which are the most profitable for the Principal. The first step towards finding potential Nash equilibria for the Agents is to be able to characterizethe so-called reaction function of the Agents. We therefore start by solving the latter problem, for i = 1 , . . . , N , and for given contract ξ and admissible actions of the other Agent a − i ∈ M N,N − .For the time being, we remain rather vague concerning the admissibility conditions for thecontracts, since these will appear naturally later on. The following arguments can therefore beseen as heuristic in this respect.Let us first define the value function of the an Agent i by U i ( a − i , ξ i ) := sup a ∈A i ( a − i ) E P a ⊗ ia − i (cid:20) − exp (cid:18) − R iA (cid:18) ξ i + Γ i ( X T ) − Z T k i ( s, a s , X s ) ds (cid:19)(cid:19)(cid:21) . The dynamic version of this stochastic control problem is then given, for any t ∈ [0 , T ] by U it ( a − i , ξ ) := ess sup a ∈A i ( a − i ) E P a ⊗ ia − i h − e − R iA ( ξ i +Γ i ( X T ) − R Tt k i ( s,a s ,X s ) ds ) (cid:12)(cid:12)(cid:12) F t i . Define then for any a ∈ A i ( a − i ) U it ( a, a − i , ξ i ) := E P a ⊗ ia − i h − e − R iA ( ξ i +Γ i ( X T ) − R Tt k i ( s,a s ,X s ) ds ) (cid:12)(cid:12)(cid:12) F t i . Then, e R iA R t k ( s,a s ,X s ) ds U it ( a, a − i , ξ ) should be an ( F , P a ⊗ i a − i ) -martingale. By the martingalerepresentation property , there should therefore exist an R N - valued and F -predictable process To be perfectly rigorous, the representation property has to be applied for the measure P , since there isno reason why in general P a should still satisfy it. This means that one has to use Bayes formula to express U it ( a, a − i , ξ ) as a conditional expectation under P and then apply the representation property. We thank SaïdHamadène for pointing out that subtlety to us. Z i,a,a − i ,ξ i such that, after applying Itô’s formula U it ( a, a − i , ξ i ) = − e − R iA ( ξ i +Γ i ( X T ) + Z Tt R iA U is ( a, a − i , ξ i ) k i ( s, a s , X s ) ds − Z Tt e − R A R s k i ( u,a u ,X u ) du e Z i,a,a − i ,ξ i s · Σ s dW a ⊗ i a − i s , ≤ t ≤ T, a.s.
By definition of W a ⊗ i a − i , we deduce that, for any t ∈ [0 , T ] , U it ( a, a − i , ξ i ) = − e − R iA ( ξ i +Γ i ( X T ) + Z Tt R iA U is ( a, a − i , ξ i ) Z i,a,a − i ,ξ i s · Σ s dW s − Z Tt R iA U is ( a, a − i , ξ i ) (cid:16) b ( s, a s ⊗ i a − is , X s ) · Z i,a,a − i ,ξ i s − k i ( s, a s , X s ) (cid:17) ds, a.s., where we have introduced the notation Z i,a,a − i ,ξ i t := − e − R iA R t k i ( s,a s ,X s ) ds R iA U it ( a, a − i , ξ i ) e Z i,a,a − i ,ξ i t , dt × d P − a.e. Then, if we set Y i,a,a − i ,ξ i t := − log (cid:0) − U it ( a, a − i , ξ i ) (cid:1) R iA , t ∈ [0 , T ] , a.s., we deduce by Itô’s formula (remember that U i ( a, a − i , ξ i ) is positive by definition) that for any t ∈ [0 , T ] , a.s. , Y i,a,a − i ,ξ i t = ξ i + Γ i ( X T ) − Z Tt Z i,a,a − i ,ξ i s · Σ s dW s + Z Tt (cid:18) − R iA (cid:13)(cid:13)(cid:13) Σ s Z i,a,a − i ,ξ i s (cid:13)(cid:13)(cid:13) + b ( s, a s ⊗ i a − is , X s ) · Z i,a,a − i ,ξ i s − k i ( s, a s , X s ) (cid:19) ds. The above equation can be identified as a linear-quadratic backward SDE with terminal condition ξ i + Γ i ( X T ) and generator ˜ f i,a − i : [0 , T ] × Ω × R N × R N −→ R , defined for any ( t, ω, z, a ) ∈ [0 , T ] × Ω × R N × R N by ˜ f i,a − i ( t, ω, z, a ) := − R iA k Σ t z k + b ( t, a ⊗ i a − it ( ω ) , X t ( ω )) · z − k i ( t, a, X t ( ω )) . Define then, for any ( t, ω, z ) ∈ [0 , T ] × Ω × R N , the map f i,a − i : [0 , T ] × Ω × R N −→ R , by f i,a − i ( t, ω, z ) := − R iA k Σ t z k + sup a ∈A i ( a − i ) (cid:8) b ( t, a ⊗ i a − it ( ω ) , X t ( ω )) · z − k i ( t, a, X t ( ω )) (cid:9) . Assume then that the BSDE with terminal condition ξ i + Γ i ( X T ) and generator f i,a − i admits amaximal solution ( Y i,a − i ,ξ i , Z i,a − i ,ξ i ) , that is to say for any t ∈ [0 , T ] Y i,a − i ,ξ i t = ξ i + Γ i ( X T ) + Z Tt f i,a − i (cid:16) s, Z i,a − i ,ξ i s (cid:17) ds − Z Tt Z i,a − i ,ξ i s · Σ s dW s , a.s., (4.1)and for any ( ˜ Y , ˜ Z ) satisfying (4.1), we have Y i,a − i ,ξ i · ≥ ˜ Y · , a.s. f i,a − i is always attained by Assumptions 2.2 and 2.3, wededuce immediately that we actually have Y i,a − i ,ξ i t = ess sup a ∈A i ( a − i ) Y i,a,a − i ,ξ i t , t ∈ [0 , T ] , a.s., which implies in turn that U it ( a − i , ξ i ) = − exp (cid:16) − R iA Y i,a − i ,ξ i t (cid:17) , t ∈ [0 , T ] , a.s., and that the optimal action of Agent i , given a contract ξ i and actions a − i of the other Agents,is given by any a ∗ ,i,a − i ,ξ i t ∈ argmax a ∈A i ( a − i ) ˜ f i,a − i (cid:16) t, Z i,a − i ,ξ i t , a (cid:17) , dt × d P − a.e. Of course, in order for our previous argumentation to be truly meaningful, the BSDEs appearingabove have to admit a maximal solution and to satisfy a comparison theorem. In order to discussthese questions, let us first establish the following lemma
Lemma 4.1.
Let Assumptions 2.2 and 2.3 hold. Then, we have for some constant
C > | f i,a − i ( t, z ) | ≤ C (cid:16) (cid:13)(cid:13) a − i (cid:13)(cid:13) + k z k (cid:17) , and any maximizer a ⋆ in the definition of f i,a − i satisfies k a ⋆ ( t, z ) k ≤ C (cid:16) k z k ℓ − (cid:17) . Proof.
First of all, notice that for any ( t, z ) ∈ [0 , T ] × R N f i,a − i ( t, z ) := − R iA k Σ t z k + sup a ∈A i ( a − i ) N X j =1 b j ( t, ( a ⊗ i a − it ) : ,j , X t ) z j − k i ( t, a, X t ) . Then, we have sup a ∈A i ( a − i ) N X j =1 b j ( t, ( a ⊗ i a − it ) : ,j , X t ) z j − k i ( t, a, X t ) ≤ sup a ∈ R N N X j =1 b j ( t, ( a ⊗ i a − it ) : ,j , X t ) z j − k i ( t, a, X t ) = N X j =1 b j (cid:0) t, ( a ⋆ ( t, z, X t ) ⊗ i a − it ) : ,j , X t (cid:1) z j − k i ( t, a ⋆ ( t, z, X t ) , X t ) , where a ⋆ ( t, z, X t ) verifies the first-order conditions ∂b j ∂a j ( t, ( a ⋆ ( t, z, X t ) ⊗ i a − it ) : ,j , X t ) z j = ∂k i ∂a j ( t, a ⋆ ( t, z, X t ) , X t ) , j = 1 , . . . , N. ∇ b j and ∇ k i ( t, a ⋆ ( t, z, X t ) , X t ) / k a ⋆ ( t, z, X t ) k ℓ − are bounded according to Assumptions2.2 and 2.3, we deduce immediately that, for some constant C > k a ⋆ ( t, z, X t ) k ≤ C (cid:16) k z k ℓ − (cid:17) . Finally, since ℓ ≥ , we have ℓ/ ( ℓ − ≤ , from which the desired result follows. ✷ Hence, by Lemma 4.1, the generator of the BSDE (4.1) has quadratic growth in z . Thus, existenceof a solution is ensured by the results of [51, 6, 1] for instance, as soon as ξ i ∈ M φ ( R ) . Whethera maximal solution exists, shall a priori require more assumptions. As for the comparison result,it is less clear, but since we will only use heuristically the result of this section, we do not try toaddress this question here (see nonetheless [7, 15, 83] for related results). The heuristic reasoning of the previous section naturally leads us to think that there should bea close connection between Nash equilibria between the N Agents and the solutions (when theyexist) of the following multidimensional BSDE Y ξt = ξ + Γ( X T ) + Z Tt f ( s, Z ξs , X s ) ds − Z Tt ( Z ξs ) ⊤ Σ s dW s , a.s., (4.2)where the map f : [0 , T ] × M N ( R ) −→ R N is defined, for = 1 , . . . , N , by f i ( t, z, x ) := f i, ( a ⋆ ) : , − i ( s,z,x ) ( s, z : ,i ) , for every ( s, z ) ∈ [0 , T ] × M N ( R ) , with the matrix a ⋆ ( s, z, x ) ∈ M N ( R ) being defined, for = 1 , . . . , N , and for every ( s, z, x ) ∈ [0 , T ] × M N ( R ) × R N , by ( a ⋆ ) : ,i ( s, z, x ) ∈ argmax a ∈A i (( a ⋆ ) : , − i ) N X j =1 b j ( t, ( a ⊗ i ( a ⋆t ) : , − i ( s, z, x )) : ,j , x ) z i,j − k i ( t, a, x ) , (4.3)where it is implicitly assumed that a given maximizer has been chosen if there are more thanone.The attentive reader should have realized that Equation (4.3) defining the map a ⋆ was actuallycircular, since a ⋆ appears on both sides of the equation. In general, it is not clear at all that a ⋆ is then well-defined. We therefore need to impose an implicit assumption on the maps b and k as follows Assumption 4.1.
For every ( s, z, x ) ∈ [0 , T ] × M N ( R ) × R N , there is at least one matrix a ⋆ ( s, z, x ) in M N ( R ) such that for any ≤ i ≤ N , ( a ⋆ ) : ,i ( s, z, x ) ∈ argmax a ∈A i (( a ⋆ ) : , − i ) N X j =1 b j ( t, ( a ⊗ i ( a ⋆t ) : , − i ( s, z, x )) : ,j , x ) z i,j − k i ( t, a, x ) . We denote by A ⋆ ( s, z, x ) the set of all matrices satisfying the above equation.
22 typical example where Assumption 4.1 is satisfied, is when the map b has a linear dependencewith respect to the effort matrix a , since in this case the maximisation in (4.3) no longer dependson the other columns of a ⋆ ( s, z, x ) . For instance, one could consider the case b i ( t, a, x ) := ˜ b i ( t, x ) + a i − N X j =1 , j = i a j , which is nothing else than a more involved version of the benchmark case of Section 3.3.2.Let us now give a precise meaning to being a solution of (4.2). Let us first start by definingthe following spaces. Fix some probability measure Q equivalent to P and a finite-dimensionalnormed vector space E , with a given norm k·k E BMO( Q , R N ) will denote the space of continuous square integrable F -martingales M (under Q ), R N valued, such that k M k BMO( Q ) < + ∞ , where k M k Q ) := ess sup τ ∈T T (cid:13)(cid:13)(cid:13) E Q [ Tr [ h M i T ] − Tr [ h M i τ ] | F τ ] (cid:13)(cid:13)(cid:13) ∞ < + ∞ , where for any t ∈ [0 , T ] , T Tt is the set of F -stopping times taking their values in [ t, T ] and wherefor any p ∈ [1 , + ∞ ] , k·k p denotes the usual norm on the space L p (Ω , F , Q ) of R -valued randomvariables. H ( Q , M N ( R )) will denote the space of M N ( R ) -valued, F -predictable processes Z such that k Z k H Q ) < + ∞ , where k Z k H Q ) := (cid:13)(cid:13)(cid:13)(cid:13)Z T Z s dW s (cid:13)(cid:13)(cid:13)(cid:13) BMO( Q ) . H ( Q , E ) will denote the space of E -valued, F -predictable processes Z s.t. k Z k H ( Q ,E ) < + ∞ , where k Z k H ( Q ,E ) := E Q (cid:20)Z T k Z s k E ds (cid:21) < + ∞ . As usual, we denote by H ( Q , E ) the localized version of this space. A solution of (4.2) isthen a pair ( Y ξ , Z ξ ) such that Y is a continuous F -semimartingale satisfying (4.2), and Z ξ ∈ H ( P , M N ( R )) .Before stating the main result of this section, we need to introduce the so-called reverse Hölderinequality. Definition 4.2 (Reverse Hölder inequality) . Fix some probability measure Q equivalent to P and some p > . A positive, or negative, F -progressively measurable process P is said to satisfy R p ( Q ) if for some constant C > τ ∈T T E Q (cid:20) (cid:18) P T P τ (cid:19) p (cid:12)(cid:12)(cid:12)(cid:12) F τ (cid:21) ≤ C, a.s.
The link between existence of a Nash equilibrium between the Agents and existence of solutionsto (4.2) is given in the following theorem. 23 heorem 4.1.
Let Assumptions 2.1, 2.2, 2.3 and 4.1 hold. There is a one-to-one correspondencebetween ( i ) a Nash equilibrium a ⋆ ( ξ ) ∈ A such that for any i = 1 , . . . , N , there exists some p > suchthat (cid:16) E P a⋆ ( ξ ) h − e − R iA ( ξ i +Γ i ( X T ) − R T k i ( s, ( a ⋆s ( ξ )) : ,i ,X s ) ds ) (cid:12)(cid:12)(cid:12) F t i(cid:17) t ∈ [0 ,T ] satisfies R p ( P a ⋆ ( ξ ) ) , ( ii ) a solution ( Y, Z ) to (4.2) , such that in addition Z ∈ H ( P , M N ( R )) , the correspondence being given by, for any i = 1 , . . . , N , ds × d P − a.e. , ( a ⋆s ( ξ )) : ,i ∈ argmax a ∈A i (( a ⋆ ) : , − i ) N X j =1 b j ( s, ( a ⊗ i ( a ⋆s ) : , − i ( s, Z s , X s )) : ,j , X s ) Z i,js − k i ( s, a, X s ) . Proof. Step . We start by showing that ( i ) leads to ( ii ) . For any ≤ i ≤ N , and for any τ ∈ T ,T , let us define the following family of random variables U i ( τ, ξ ) := ess sup a ∈A i (( a ⋆ ( ξ )) : , − i ) E P a ⊗ i ( a⋆ ( ξ )): , − i h − e − R iA ( ξ i +Γ i ( X T ) − R Tτ k i ( s,a s ,X s ) ds ) (cid:12)(cid:12)(cid:12) F τ i . It is a classical result that this family satisfies the following dynamic programming principle (seefor instance Theorem 2.4 in [27] as well as the discussion in their Section 2.4.2) U i ( τ, ξ ) = ess sup a ∈A i (( a ⋆ ( ξ )) : , − i ) E P a ⊗ i ( a⋆ ( ξ )): , − i h e R iA R θτ k i ( s,a s ,X s ) U i ( θ, ξ ) (cid:12)(cid:12)(cid:12) F τ i , for any θ ∈ T ,T such that τ ≤ θ, a.s.It is then immediate that for any a ∈ A i (( a ⋆ ( ξ )) : , − i ) , the family ( e R iA R τ k i ( s,a s ,X s ) U i ( τ, ξ )) τ ∈T ,T is a so-called P a ⊗ i ( a ⋆ ( ξ )) : , − i -supermartingale system. Hence, by the results of [16], it can beaggregated by a unique (up to indistinguishability) F -optional process, which actually coincides,a.s., with ( e R iA R t k i ( s,a s ,X s ) U it (( a ⋆ ( ξ )) : , − i , ξ )) t ∈ [0 ,T ] defined in the previous section.Moreover, this aggregator remains a P a ⊗ i ( a ⋆ ( ξ )) : , − i -supermartingale, which then admits a càdlàgmodification (remember that the filtration considered satisfies the usual conditions). Let usnow check that ( e R iA R t k i ( s, ( a ⋆ ( ξ )) : ,is ,X s ) U it (( a ⋆ ( ξ )) : , − i , ξ )) t ∈ [0 ,T ] is a uniformly integrable P a ⋆ ( ξ ) -martingale.By definition of a Nash equilibria, ( a ⋆ ( ξ )) : ,i is optimal for the problem of Agent i , that is U i (( a ⋆ ( ξ )) : , − i , ξ ) = E P a⋆ ( ξ ) h − e − R iA ( ξ i +Γ i ( X T ) − R T k i ( s, ( a ⋆s ( ξ )) : ,i ,X s ) ds ) i . Next, by the supermartingale property proved above (which holds for any choice of action ofAgent i ), and by definition of the value function of Agent i , we must have U i (( a ⋆ ( ξ )) : , − i , ξ ) ≥ E P a⋆ ( ξ ) h e R iA R t k i ( s, ( a ⋆s ( ξ )) : ,i ,X s ) U it (( a ⋆ ( ξ )) : , − i , ξ ) i ≥ E P a⋆ ( ξ ) h − e − R iA ( ξ i +Γ i ( X T ) − R T k i ( s, ( a ⋆s ( ξ )) : ,i ,X s ) ds ) i = U i (( a ⋆ ( ξ )) : , − i , ξ ) , t ∈ [0 , T ] . Hence, all these terms have to be equal, which implies in particular that e R iA R t k i ( s, ( a ⋆s ( ξ )) : ,i ,X s ) U it (( a ⋆ ( ξ )) : , − i , ξ ) = E P a⋆ ( ξ ) h − e − R iA ( ξ i +Γ i ( X T ) − R T k i ( s, ( a ⋆s ( ξ )) : ,i ,X s ) ds ) (cid:12)(cid:12)(cid:12) F t i , for any t ∈ [0 , T ] . This provides the desired result, since the right-hand side is obviously a P a ⋆ ( ξ ) -martingale, as the conditional expectation of an integrable random variable, and sinceby Assumptions 2.2 and 2.3 and by definition of admissible controls, this martingale actuallybelongs to L p (Ω , F , P a ⋆ ( ξ ) ) , for any p ≥ (with moments uniformly bounded in t ∈ [0 , T ] by Doob’s inequality). Since it is also negative, by the predictable martingale representationproperty, there exists an F -predictable process Z i,a ⋆ ( ξ ) ,ξ ∈ H ( P a ⋆ ( ξ ) , R N ) such that, for any t ∈ [0 , T ] e R iA R t k i ( s, ( a ⋆s ( ξ )) : ,i ,X s ) U it (( a ⋆ ( ξ )) : , − i , ξ ) = U i (( a ⋆ ( ξ )) : , − i , ξ ) E (cid:18)Z · Z i,a ⋆ ( ξ ) ,ξs · Σ s dW a ⋆ ( ξ ) s (cid:19) t , a.s. Then, since by assumption e R iA R t k i ( s, ( a ⋆s ( ξ )) : ,i ,X s ) U it (( a ⋆ ( ξ )) : , − i , ξ ) satisfies ( R p ( P )) , we can useTheorems 3.3 and 3.4 of [50] to deduce that Z i,a ⋆ ( ξ ) ,ξ belongs to both H ( P a ⋆ ( ξ ) , R ) and H ( P , R ) . Furthermore, a simple application of Itô’s formula leads to U it (( a ⋆ ( ξ )) : , − i , ξ )= − e − R iA ( ξ i +Γ i ( X T )) + Z Tt R iA U is (( a ⋆ ( ξ )) : , − i , ξ ) Z i,a ⋆ ( ξ ) ,ξs · Σ s dW s − Z Tt R iA U is (( a ⋆ ( ξ )) : , − i , ξ ) (cid:16) b ( s, a ⋆s ( ξ ) , X s ) · Z i,a ⋆ ( ξ ) ,ξs − k i ( s, ( a ⋆s ( ξ )) : ,i , X s ) (cid:17) ds. Therefore, we deduce that for any a ∈ A i (( a ⋆ ( ξ )) : , − i ) , we have ( R iA ) − e R iA R t k i ( s,a s ) ds U it (( a ⋆ ( ξ )) : , − i , ξ )= U i (( a ⋆ ( ξ )) : , − i , ξ ) R iA − Z t e R iA R s k i ( r,a r ,X r ) dr U is (( a ⋆ ( ξ )) : , − i , ξ ) Z i,a ⋆ ( ξ ) ,ξs · Σ s dW a ⊗ i ( a ⋆ ( ξ )) : , − i s + Z t e R iA R s k i ( r,a r ,X r ) dr U is (( a ⋆ ( ξ )) : , − i , ξ )( b ( s, a ⋆s ( ξ ) , X s ) · Z i,a ⋆ ( ξ ) ,ξs − k i ( s, ( a ⋆s ( ξ )) : ,i , X s )) ds − Z t e R iA R s k i ( r,a r ,X r ) dr U is (( a ⋆ ( ξ )) : , − i , ξ )( b ( s, a s ⊗ i ( a ⋆s ( ξ )) : , − i , X s ) · Z i,a ⋆ ( ξ ) ,ξs − k i ( s, a s , X s )) ds. Now remember that this process must be a P a ⊗ i ( a ⋆ ( ξ )) : , − i -supermartingale. This therefore impliesthat we must have for any a ∈ A i (( a ⋆ ( ξ )) : , − i ) , a.s. , b ( s, a ⋆s ( ξ ) , X s ) · Z i,a ⋆ ( ξ ) ,ξs − k i ( s, ( a ⋆s ( ξ )) : ,i , X s ) ≥ b ( s, a s ⊗ i ( a ⋆s ( ξ )) : , − i , X s ) · Z i,a ⋆ ( ξ ) ,ξs − k i ( s, a s , X s ) . In other words ( a ⋆s ( ξ )) : ,i ∈ argmax a ∈A i (( a ⋆ ) : , − i ) n b ( s, a ⊗ i ( a ⋆s ( ξ )) : , − i , X s ) · Z i,a ⋆ ( ξ ) ,ξ − k i ( s, a, X s ) o . Define then Y it ( a ⋆ ( ξ ) , ξ ) := − log (cid:0) − U it (( a ⋆ ( ξ )) : , − i , ξ ) (cid:1) R iA , t ∈ [0 , T ] , a.s.
25e have immediately for any t ∈ [0 , T ] , a.s. , Y it ( a ⋆ ( ξ ) , ξ ) = ξ i + Γ i ( X T ) − Z Tt Z i,a ⋆ ( ξ ) ,ξs · Σ s dW s + Z Tt (cid:18) − R iA (cid:13)(cid:13)(cid:13) Σ s Z i,a ⋆ ( ξ ) ,ξs (cid:13)(cid:13)(cid:13) + b ( s, a ⋆s ( ξ ) , X s ) · Z i,a ⋆ ( ξ ) ,ξs − k i ( s, ( a ⋆s ) : ,i , X s ) (cid:19) ds. Since all of this has to hold true for any i = 1 , . . . , N , we deduce that if we define Y t ( a ⋆ ( ξ ) , ξ ) as the vector in R N whose i th coordinate is Y it ( a ⋆ ( ξ ) , ξ ) , and if we define Z a ⋆ ( ξ ) ,ξ as the N × N matrix whose i th column is Z i,a ⋆ ( ξ ) ,ξ , then the pair ( Y t ( a ⋆ ( ξ ) , ξ ) , Z a ⋆ ( ξ ) ,ξ ) is a solution of theBSDE (4.2), such that Z a ⋆ ( ξ ) ,ξ ∈ H ( P , M N ( R )) . Step . Conversely, let us be given a solution ( Y, Z ) to (4.2) s.t. Z ∈ H ( P , M N ( R )) .A classical measurable selection argument allows us to define a M N ( R ) -valued F -predictableprocess a ⋆ such that for any i = 1 , . . . , N ( a ⋆s ) : ,i ∈ argmax a ∈A i (( a ⋆ ) : , − i ) (cid:8) b ( s, a ⊗ i ( a ⋆s ) : , − i ( Z s , X s ) , X s ) · Z : ,is − k i ( s, a, X s ) (cid:9) . Define then for any i = 1 , . . . , N U it := − exp (cid:0) − R iA Y it (cid:1) , a.s. We can then go backwards in all the computations of Step , to obtain that, thanks to thefact that Z ∈ H ( P , M N ( R )) , ( e R iA R t k i ( s,a s ,X s ) U it ) t ∈ [0 ,T ] is a P a ⊗ i ( a ⋆ ) : , − i -supermartingale forany a ∈ A i (( a ⋆ ) : , − i ) , and that ( e R iA R t k i ( s, ( a ⋆s ) : ,i ,X s ) U it ) t ∈ [0 ,T ] is a P a ⋆ -martingale. This uses inparticular the fact that the Doléans-Dade exponential of a BMO-martingale is a uniformlyintegrable martingale. By the martingale optimality principle, this implies that U i = U i (( a ⋆ ) : , − i , ξ ) , and that ( a ⋆ ) : ,i is optimal for the problem of Agent i . Since this holds for any i = 1 , . . . , N , thismeans that a ⋆ is a Nash equilibrium.Finally we have to check that the a ⋆ we have defined is such that for any i = 1 , . . . , N , thereexists some p > such that (cid:16) E P a⋆ h − e − R iA ( ξ i +Γ i ( X T ) − R T k i ( s, ( a ⋆s ) : ,i ,X s ) ds ) (cid:12)(cid:12)(cid:12) F t i(cid:17) t ∈ [0 ,T ] satisfies R p ( P a ⋆ ) .But − e − R iA ( ξ i +Γ i ( X T ) − R T k i ( s, ( a ⋆s ) : ,i ,X s ) ds ) = e R iA R T k i ( s, ( a ⋆s ) : ,i ,X s ) ds U iT , so that for any t ∈ [0 , T ] X it := E P a⋆ h − e − R iA ( ξ i +Γ i ( X T ) − R T k i ( s, ( a ⋆s ) : ,i ,X s ) ds ) (cid:12)(cid:12)(cid:12) F t i = e R iA R t k i ( s, ( a ⋆s ) : ,i ,X s ) ds U it = U i E (cid:18)Z · Z is · Σ s dW a ⋆ s (cid:19) t . Hence, since Z ∈ H ( P , M N ( R )) , we deduce from Theorem 3.4 in [50] that the desired resultholds. ✷ .1.4 On existence of Nash equilibria and admissible contracts The main result of the previous section gives a complete characterization of the Nash equilibriafor the Agents, which satisfy some integrability conditions, as solutions to the multidimensionalquadratic BSDE (4.2). However, this does not address the question of existence of these equilib-ria, and this is exactly where the heart of the problem lies. Indeed, unlike in the one-dimensionalcase (that is N = 1 ) wellposedness results for this kind of BSDE are extremely scarce in theliterature. Tevzadze [83] was the first to obtain a wellposedness result in the case of a boundedand sufficiently small terminal condition. It was then proved by Frei and dos Reis [35] and Frei[34] that even in seemingly benign situations, existence of global solutions could fail. Later on,Cheredito and Nam [9], Kardaras et al. [49], Kramkov and Pulido [54, 55], Hu and Tang [45],Jamneshan et al. [47], or more recently Luo and Tangpi [57] all obtained some positive results,but only in particular instances, which do not readily apply in our setting. Recently, Xingand Žitković [86] obtained quite general existence and uniqueness results, but in a Markovianframework.Therefore, since the Principal wants to offer contracts which reveal the actions of the Agents,he will never offer a contract for which a Nash equilibria between the Agents does not exist. Forsimplicity, let us introduce, for any ξ ∈ C F B , the set
NA( ξ ) of Nash equilibria associated to ξ and which satisfy condition ( i ) in Theorem 4.1 (which can be the empty set according to theabove discussion). Furthermore, we also define NAI( ξ ) := { a ∈ NA( ξ ) , a (cid:23) b, for any b ∈ NA( ξ ) } . Besides, we remind the reader that, in line with the classical literature on the subject, we assumethat since the Agents are indifferent between Nash equilibria in
NAI( ξ ) , the Principal can makethem choose the one that suits him best.This motivates the following definition of admissible contracts for the second-best problem C SB := (cid:8) ξ ∈ C F B , NAI( ξ ) is non-empty (cid:9) . As a consequence of Theorem 4.1, we know that for any ξ ∈ C SB , there exists a pair ( Y ξ , Z ξ ) ∈ R N × H ( P , M N ( R )) such that ξ = Y ξ − Γ( X T ) − Z T f ( s, Z ξs , X s ) ds + Z T ( Z ξs ) ⊤ Σ s dW s , a.s. (4.4)where, the optimal feedback control a ⋆ being given by ???, we recall that the vector function f is defined by f i : ( t, z, x )
7→ − R iA (cid:13)(cid:13) Σ t z : ,i (cid:13)(cid:13) − k i ( t, ( a : ,i ) ⋆ ( t, z, x ) , x ) + N X j =1 b j ( t, ( a : ,j ) ⋆ ( t, z, x ) , x ) z j,i . For any ( Y , Z ) ∈ R N × H ( P , M N ( R )) , define ξ Y ,Z := Y − Γ( X T ) − Z T f ( s, Z s , X s ) ds + Z T ( Z s ) ⊤ Σ s dW s .
27y (4.4), we know that the set of admissible contracts C SB is actually included in the set (cid:8) ξ Y ,Z , ( Y , Z ) ∈ R N × H ( P , M N ( R )) (cid:9) . As usual in Holmström-Milgrom type problems, we know that the value of the constant vector Y in the contract will be fine-tuned so that each Agent receives exactly his reservation utility.But this exactly corresponds to choosing Y i = L i := − ln( − U i ) /R iA , ≤ i ≤ N. We can therefore consider the following problem, which is, a priori, an upper bound of thePrincipal value function U P,SB := sup Z ∈ H ( P , M N ( R )) sup a ∈ NAI( ξ L,Z ) E P a h − e − R P ( X T − ξ L,Z ) · N i . (4.5)Then, under this form, we can interpret ξ L,Z := ξ L,Z + Γ( X T ) as the terminal value of thefollowing Markovian controlled diffusion Y t = L − Z t f ( s, Z s , X s ) ds + Z t ( Z s ) ⊤ Σ s dW s , t ∈ [0 , T ] , where the control process is actually Z . Furthermore, for simplicity of notations, we assumethat the maximizer in the definition of f is actually unique. We are thus back into the realm ofclassical stochastic control, and, at least formally, we can identify U P,SB with the value v (0 , , L ) of the unique solution in an appropriate sense to the following HJB equation ( ( v t + G ( · , , v x , v y , v xx , v yy , v xy )) ( t, x, y ) = 0 , ( t, x, y ) ∈ [0 , T ) × R N ,v ( T, x, y ) = − exp ( − R P ( x + Γ( x ) − y ) · N ) , ( x, y ) ∈ R N , (4.6)where G ( t, x, p, q, γ, η, ν ) := sup z ∈M N ( R ) (cid:26) b ( t, a ⋆ ( t, z, x ) , x ) · p − f ( t, z, x ) · q + 12 Tr h Σ t Σ ⊤ t γ i + 12 Tr h z ⊤ Σ t Σ ⊤ t zη i + Tr h Σ t Σ ⊤ t zν i(cid:27) . The following result follows from a classical verification argument, see for instance [33], so thatwe will not provide its proof.
Proposition 4.1.
Assume that the PDE (4.6) admits a unique classical ( that is C , , ) solution,and that the supremum in the definition of G is attained for at least one z ⋆ ( t, x, p, q, γ, η, ν ) .Assume that z ⋆t ∈ H ( P , M N ( R ) where z ⋆t := z ⋆ ( t, X ⋆t , v x ( t, X ⋆t , Y ⋆t ) , v y ( t, X ⋆t , Y ⋆t ) , v xx ( t, X ⋆t , Y ⋆t ) , v yy ( t, X ⋆t , Y ⋆t ) , v xy ( t, X ⋆t , Y ⋆t )) , where X ⋆ and Y ⋆ are the unique solutions ( assumed to exist ) of the coupled SDEs X ⋆t = Z t b ( s, a ⋆ ( s, z ⋆s , X ⋆s ) , X ⋆s ) ds + Z t Σ s dW s ,Y ⋆t = Y − Z t f ( s, z ⋆s , X ⋆s ) ds + Z t ( z ⋆s ) ⊤ Σ s dW s . Assume furthermore that for any Y ∈ R N , the contract ξ Y ,z ⋆ ∈ C SB . Then, the value functionof the Principal is given by v (0 , , L ) . Of course uniqueness needs to be verified in practice
28f course, as usual with verification type results, the above proposition is a bit disappointing.This is the reason why we consider in the next section a more specific problem for which theproblem of the Principal can actually be solved diretly, without having to refer to the HJBequation (4.6). This particular case consists in considering a linear comparison function Γ as inSection 3.3. Γ We now focus on the particular case where γ is linear and each agent compares the terminalvalue of his project to the average of those of all the projects. Namely, we again suppose as inSection 3.3 that Assumption 3.4 holds. The Principal problem can be written as U P,SB = sup ξ ∈C SB sup a ∈ NAI( ξ ) E P a " − e − R P ( X T − ξ ) · N − N X i =1 ρ i e − R iA (cid:16) ξ i + γ i (cid:16) X iT − X − iT (cid:17) − R T k i ( s,a : ,is ) ds (cid:17) , (4.7)where the Lagrange multipliers ( ρ i ) ≤ i ≤ N are once again here to ensure the participation con-straints of the Agents. Plugging (4.4) in (4.7), we deduce U P,SB = sup ξ ∈C SB sup a ∈ NAI( ξ ) E P a h − E (cid:16) Z · (cid:16) ⊤ N Z ξs + ⊤ N + γ ⊤ − ( γ − ) ⊤ (cid:17) Σ s dW as (cid:17) T e R P Y ξ · N × e − R p R T β ( s,Z ξs ) ds − N X i =1 ρ i e − R iA ( Y ξ ) i E (cid:16) − R iA Z · (( Z ξs ) ⊤ Σ s dW as ) i (cid:17) T i , where the map β : [0 , T ] × M N ( R ) −→ R is defined by β ( t, z ) := (cid:0) N + γ − γ − (cid:1) · b ( t, a ( z )) − k ( t, a ( z )) · N − N X i =1 R iA (cid:13)(cid:13) Σ t z : ,i (cid:13)(cid:13) − R P (cid:13)(cid:13)(cid:13) Σ t (cid:16) z ⊤ N + N + γ − γ − (cid:17)(cid:13)(cid:13)(cid:13) . Now it is clear by Assumptions 2.2, 2.3 and Lemma 4.1 that the map β is continuous in z and goes to −∞ as k z k goes to ∞ , so that it admits at least one deterministic maximizer,which we denote by z ⋆t . Then, recalling that Z ξ belongs to H ( P , M N ( R )) and thus also to H ( P a , M N ( R )) for any a ∈ A by Theorem 3.3 in [50], we deduce that U P,SB ≤ sup Y ξ ∈ R N ( − e R P Y ξ · N − R T β ( s,z ⋆s ) ds − N X i =1 ρ i e − R iA ( Y ξ ) i ) . The right-hand side is a concave function of Y ξ on R N , whose supremum is attained at Y ⋆ , with ( Y ⋆ ) i := 1 R iA log (cid:18) ρ i R iA R P (cid:19) − R P R A R iA (cid:0) R A + N R P (cid:1) N X j =1 R jA log ρ j R jA R P ! + R A R iA (cid:0) R A + N R P (cid:1) Z T β ( s, z ⋆s ) ds, i = 1 , . . . , N, and we directly deduce that U P,SB ≤ − R A + N R P R A e RARA + NRP (cid:18)P Ni =1 1 RiA log (cid:18) ρiRiARP (cid:19) − R T β ( s,z ⋆s ) ds (cid:19) . (4.8)29n order to attain this upper bound, we would like to consider the contract ξ ⋆ defined by ξ ⋆ := Y ⋆ − diag( γ )( X T − X − T ) − Z T f ( s, z ⋆s ) ds + Z T z ⋆s Σ s dW s . But under this form, it is clear that the corresponding BSDE (4.2) admits at least as a solution ( Y ⋆ , z ⋆ ) , where Y ⋆t := Y ⋆ − Z t f ( s, z ⋆s ) ds + Z t ( z ⋆s ) ⊤ Σ s dW s , t ∈ [0 , T ] , a.s. Moreover, since z ⋆ is deterministic, it is clearly in H ( P , M N ( R )) . Hence, we do have ξ ⋆ ∈C SB , so that equality holds in (4.8).Finally, it remains to choose the Lagrange multipliers so as to satisfy the participation constraintsof the Agents, with reservation utilities ( U i ) ≤ i ≤ N . We must therefore solve − e − R iA ( Y ⋆ ) i = U i , i = 1 , . . . , N. Arguing exactly as in the first-best case, we compute that ρ i = − R P U i R iA N Y j =1 ( − U j ) RPRjA − e − R T β ( s,z ⋆s ) ds . We have thus proved
Theorem 4.2.
Let Assumptions 2.1, 2.2, 2.3, 3.4 and 4.1 hold. Then an optimal contract ξ SB ∈ C SB in the problem (4.8) , with reservation utilities ( U i ) ≤ i ≤ N ∈ ( −∞ , N , is given, for i = 1 , . . . , N , by ξ iSB := − R iA log( − U i ) − γ i ( X iT − X − iT ) + (cid:18)Z T z ⋆s dX s (cid:19) i + Z T k i ( s, ( a ⋆s (( z ⋆s ) i, : )) : ,i ) ds + R iA Z T (cid:13)(cid:13) Σ( z ⋆s ) : ,i (cid:13)(cid:13) ds, where the matrix a ⋆s ( z ) ∈ M N ( R ) is defined implicitly , for every ( s, z ) ∈ [0 , T ] × M N ( R ) andfor = 1 , . . . , N , by ( a ⋆s ( z )) : ,i ∈ argmax a ∈A i (( a ⋆ ) : , − i ) (cid:8) b ( t, a ⊗ i ( a ⋆s ( z )) : , − i ) · z : ,i − k i ( t, a ) (cid:9) , and where z ⋆t is any maximizer of the map z (cid:0) N + γ − γ − (cid:1) · b ( t, a ⋆t ( z )) − k ( t, a ⋆t ( z )) · N − N X i =1 R iA (cid:13)(cid:13) Σ t z : ,i (cid:13)(cid:13) − R P (cid:13)(cid:13)(cid:13) Σ t (cid:16) z ⊤ N + N + γ − γ − (cid:17)(cid:13)(cid:13)(cid:13) . .2.3 Back to the bidimensional linear-quadratic benchmark setting We once again focus on the benchmark case developed in Section 3.3.2 for which explicit com-putations are available. In this framework, let’s recall that b ( t, a ) := a , − a , a , − a , ! , for any a := a , a , a , a , ! ∈ M ( R ) , and for some constants ( k , , k , , k , , k , ) ∈ ( R ⋆ + ) k ( t, a ) := k , (cid:12)(cid:12) a , (cid:12)(cid:12) + k , (cid:12)(cid:12) a , (cid:12)(cid:12) k , (cid:12)(cid:12) a , (cid:12)(cid:12) + k , (cid:12)(cid:12) a , (cid:12)(cid:12) ! , for any a := a , a , a , a , ! ∈ M ( R ) . Easy computations show that the optimal control matrix a ⋆ ( z ) is given by a ⋆ ( z ) = z , k , z , k , z , k , z , k , ! . To find the optimal value z ⋆ , we therefore need to maximize the map g : z (1 + γ − γ ) (cid:18) z , k , − z , k , (cid:19) + (1 + γ − γ ) (cid:18) z , k , − z , k , (cid:19) − (cid:12)(cid:12) z , (cid:12)(cid:12) k , − (cid:12)(cid:12) z , (cid:12)(cid:12) k , − (cid:12)(cid:12) z , (cid:12)(cid:12) k , − (cid:12)(cid:12) z , (cid:12)(cid:12) k , − σ (cid:16) R A (cid:12)(cid:12) z , (cid:12)(cid:12) + R A (cid:12)(cid:12) z , (cid:12)(cid:12) (cid:17) − σ (cid:16) R A (cid:12)(cid:12) z , (cid:12)(cid:12) + R A (cid:12)(cid:12) z , (cid:12)(cid:12) (cid:17) − R P σ (cid:0) z , + z , + 1 + γ − γ (cid:1) − R P σ (cid:0) z , + z , + 1 + γ − γ (cid:1) . Easy (but lengthy) calculations show that g is actually concave, and admits a unique criticalpoint given by, for i, j = 1 , z ⋆ ) i,i = 1 α i,j (cid:16) (1 + k i,i k i,j R p R iA σ σ )(1 + γ i − γ j ) + 2 k i,i k i,j (cid:12)(cid:12) R p σ i (cid:12)(cid:12) (cid:17) ( z ⋆ ) j,i = − α i,j , (cid:0) (2 + σ i k i,i R iA )(1 + γ i − γ j ) + R p σ i k i,j (1 + σ i k i,i ( R iA + R p ))(1 + γ j − γ i ) (cid:1) , where α i,j := 1 + σ i ( R iA + R P ) k i,i + ( σ j R iA + σ i R P ) k j,i + σ i R iA ( σ j ( R iA + R P ) + σ i R P ) k j,i k i,i . Whenever γ is significantly higher than γ , observe that once again Agent optimal strategycan be to work against its own project. As in the first best case developed above, observe that the optimal contract is linear. Moreovereach agent obtains his reservation utility and is paid a given proportion of the value of eachproject. As observed explicitly in the linear-quadratic benchmark case, the analytic form of theoptimal contract is more intricate than in the first best setting, but most qualitative properties ofthe optimal contract and strategies remain unchanged. In particular, each agent has incentives31o help the project of a very competitive colleague, whereas he may have to work against theproject of a poorly competitive one. In some sense, a competitive agent rewards himself via hisappetite for competition, and therefore it is in the economic interest of the Principal to providehim a project with higher probability of success, e.g. with less volatility or the help of the otherAgents.We recall that we assumed throughout the paper that the competition indexes γ of candidatesare observable for the Principal. In such framework, we emphasize that our results imply inparticular that it is not optimal for the Principal to hire Agents with similar appetence forcompetition. Economic benefits for the firm (i.e. the Principal) follows from a diversificationof competitive profiles, while each Agent always recovers his reservation utility. A pertinentfuture research topic may be to inquire if such conclusions remain valid, in an adverse selectionframework, where the competitive appetence of the agent are ex-ante unknown, and shall bedetermined via the design of a well suited menu of contracts. References [1] Barrieu, P., El Karoui, N. (2013). Monotone stability of quadratic semimartingales with applica-tions to unbounded general quadratic BSDEs,
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