Asymptotic efficiency of the proportional compensation scheme for a large number of producers
aa r X i v : . [ q -f i n . E C ] J a n ASYMPTOTIC EFFICIENCY OF THE PROPORTIONALCOMPENSATION SCHEME FOR A LARGE NUMBER OF PRODUCERS
DMITRY B. ROKHLIN AND ANATOLY USOV
Abstract.
We consider a manager, who allocates some fixed total payment amount be-tween N rational agents in order to maximize the aggregate production. The profit of i -thagent is the difference between the compensation (reward) obtained from the manager andthe production cost. We compare (i) the normative compensation scheme, where the man-ager enforces the agents to follow an optimal cooperative strategy; (ii) the linear piece rates compensation scheme, where the manager announces an optimal reward per unit good; (iii)the proportional compensation scheme, where agent’s reward is proportional to his contri-bution to the total output. Denoting the correspondent total production levels by s ∗ , ˆ s and s respectively, where the last one is related to the unique Nash equilibrium, we examine thelimits of the prices of anarchy A N = s ∗ /s , A ′ N = ˆ s/s as N → ∞ . These limits are calculatedfor the cases of identical convex costs with power asymptotics at the origin, and for powercosts, corresponding to the Coob-Douglas and generalized CES production functions withdecreasing returns to scale. Our results show that asymptotically no performance is lost interms of A ′ N , and in terms of A N the loss does not exceed 31%. Introduction
Consider a manager, who allocates some fixed some fixed total payment amount M be-tween N producers (agents) in order to maximize the aggregate production. The profit ofeach agent equals to the difference between the reward obtained from the manager and theproduction cost ϕ i . If the cost functions are known, the manager can determine rewards,stimulating the optimal aggregate production s ∗ . We call such compensation scheme nor-mative . Besides the quite unrealistic assumption that the cost functions are known, thisscheme suffers from another drawback: it does not announce any common reward sharingrules. However, s ∗ can serve as a benchmark.Another idea is to use the linear piece rates compensation scheme, announcing a price µ of the unit good. So, the reward µx i of i -th agent will be linear in his production level x i . Assuming an individually optimal (rational) agent behaviour, the manager can chose µ in such a way that the total reward does not exceed M , and the total production ˆ s cannotbe improved by another linear reward rule. Clearly, this compensation scheme also requiresthe knowledge of production cost functions, although it is easier to assign one parameter µ , rather than the full set of rewards, as in the normative scheme. Note also that in thepiece rates allocation scheme the total reward requested by irrational agents can exceed M .Nevertheless, we regard the value ˆ s as another benchmark.The main focus of the present study is the proportional compensation scheme, where thereward M x i / ( x + · · · + x N ) of i -th agent is proportional to his contribution to the aggregateproduction. The realization of this scheme requires no information concerning the cost Mathematics Subject Classification.
Key words and phrases.
Proportional compensation scheme, total production, price of anarchy, asymptoticefficiency, Tullock contest. functions, and the total reward equals to M irrespective of agent actions (except the trivialcase, where x = 0). So, the manager allows the agents to determine optimal productionlevels on their own in the course of a (non-cooperative) game with the payoff functions M x i x + · · · + x N − ϕ i ( x i ) . (1.1)Under the assumption that the cost functions ϕ i are convex and strictly increasing, the game(1.1) has a unique Nash equilibrium. By s we denote the correspondent total production.One may argue that the computation of a Nash equilibrium also requires the knowledgeof cost functions. However, such equilibrium also can emerge as a result of agent interactionin a repeated game through the mechanism of no-regret learning. We recall this concept atthe end of the paper. For each agent the no-regret learning does not require the knowledgeof the cost functions of other agents.The game (1.1) is a special case of the Cournot oligopoly: [27, 25], and it fits into theextensively studied theory of contests : see [7, 21, 8, 32] for reviews (an experimental researchis reviewed in [12]). In a contest the payoff function of each player is the difference betweenthe contest success function (CSF) and the cost of player’s effort. A player’s CSF usuallyequals to the expected value of winning an indivisible prize, or, as in our case, to the portionof the prize, obtained by the player. It depends on the efforts of all players, and it isincreasing in the effort of a selected player and decreasing in the efforts of the other ones.An account of the CSF’s can be found in [18].Using the substitution f i ( y i ) = ϕ − i ( y i ), we can reduce the game (1.1) to a strategicallyequivalent contest M f i ( y i ) f ( y ) + · · · + f N ( y N ) − y i (1.2)with the CSF of the general-logit form (in the terminology of [32, Chapter 4]). In addition,in our main example of power costs ϕ i = c i x αi , α ≥
1, corresponding to generalized CESproduction functions with decreasing returns to scale, the game (1.2) boils down to the
Tullock contest with f i ( y i ) = ( y i /c i ) /α .Contests are used to model conflict situations in rent-seeking, resource allocation, patentraces, sports, advertising, etc. The present paper is related to the analysis of relative perfor-mance incentive schemes in labour contracts. An active study of such problems was initiatedin 1980s: [23, 14, 26, 24]. We also mention several recent papers with an emphasis on ex-perimental and empirical studies: [6, 15, 29], where the reader can find a lot of additionalreferences. The prevailing concept is the rank-order allocation of prizes (rank-order tourna-ments). However, the proportional prize-contest was promoted by the means of experimentalstudies in [3].The main feature of the present paper is the analysis of the following two versions of the“price of anarchy”: A N = s ∗ /s, A ′ N = ˆ s/s for a large number N of agents. In line with [22]the price of anarchy shows how much performance is lost by the lack of coordination. Thestudy of the prices of anarchy recently became an active area of research. We mention onlya few papers, studying an efficiency of the proportional resource allocation mechanism insomewhat different models: [19, 5, 2].In Section 2 we describe three compensation schemes mentioned above. In particular, wepoint out that any contest scheme cannot be better than the normative one (Remark 2). InSection 3 we study the prices of anarchy A N , A ′ N for large N . Our results show that for thecases of identical convex costs ϕ i = ϕ with power asymptotics at the origin (Theorem 1), and SYMPTOTIC EFFICIENCY OF THE PROPORTIONAL COMPENSATION SCHEME 3 for heterogeneous agents with power costs ϕ i ( x ) = c i x α , α > A ′ N , and in terms of A N the loss does not exceed 31%.These results characterize an asymptotic efficiency of the proportional compensation scheme.We also conjecture that this result remains true for heterogeneous agents with linear costfunctions ( α = 1) and i.i.d. marginal costs c i .2. Three compensation schemes
Let x i be the amount of good produced by i -th agent. Denote by ϕ i : R + R + the relatedproduction cost. We assume that the functions ϕ i are twice continuously differentiable, ϕ i (0) = 0, ϕ ′ i ( x i ) > ϕ ′′ i ( x i ) ≥ x i >
0. It easily follows that ϕ i ( x i ) → + ∞ , x i → + ∞ .(i) Normative compensation scheme . Agent i knows the reward function ψ i ( x i ) ≥ ψ i ( x i ) − ϕ ( x i ) → max x i ≥ . (2.1)Let ψ i (0) = 0, and denote by ˜ x i = ˜ x i ( ψ i ) optimal solutions of (2.1), which for simplicity weassume to exist. The manager has M units of capital at his disposal. His aim is to maximizethe total production: N X i =1 ˜ x i → maxover all reward functions ψ i , satisfying the conditions N X i =1 ψ i (˜ x i ) ≤ M ; ψ i ≥ , i = 1 , . . . , N. Since ψ i (˜ x i ) − ϕ i (˜ x i ) ≥
0, we get the estimate N X i =1 ϕ i (˜ x i ) ≤ N X i =1 ψ i (˜ x i ) ≤ M. Thus, given the budget M , the total production cannot exceed the value s ∗ = sup ( N X i =1 x i : N X i =1 ϕ i ( x i ) ≤ M, x ≥ ) (2.2)for any kind of rewards ψ i .On the other hand, it is possible to obtain the total production arbitrary close to s ∗ byannouncing the rewards ψ i ( x i ) = ϕ i ( x ∗ i ) I [ x ∗ i − ε i , ∞ ) ( x i ) , (2.3)where x ∗ = ( x ∗ i ) Ni =1 is an optimal solution of (2.2) and ε i ∈ (0 , x ∗ i ) , if x ∗ i > ε i = 0 , if x ∗ i = 0 . Indeed, in this case the optimal solution of (2.1) is of the form˜ x i = ( x ∗ i − ε i , x ∗ i > , , x ∗ i = 0and P Ni =1 ˜ x i = s ∗ − P Ni =1 ε i , while P Ni =1 ψ i (˜ x i ) = P Ni =1 ϕ i ( x ∗ i ) ≤ M. Thus, one can regard s ∗ as the optimal total production amount under the normative compensation scheme. DMITRY B. ROKHLIN AND ANATOLY USOV (ii)
Linear piece rates compensation scheme . Assume that the cost functions are strictlyconvex and the manager tries to choose a best linear reward function ψ i ( x i ) = µx i . Theproduction levels ˆ x i ( µ ) are determined by the problems µx i − ϕ i ( x i ) → max x i ≥ . (2.4)The functions ˆ x i ( µ ) are non-decreasing, and the best choice of µ corresponds to the largesttotal production which does not violate the budget constraint: µ N X i =1 ˆ x i = M. (2.5)The aggregate production is given by ˆ s = P Ni =1 ˆ x i . (iii) Proportional compensation scheme . The left-hand side of (2.5) equals to the totalreward. If the agents anticipate that the manager selects µ in this way, then they becomeinvolved in the non-cooperative game with the payoff functions H i ( x ) = M x i P Ni = j x j − ϕ i ( x i ) , x ≥ / s = N X i =1 x i , where x is the unique Nash equilibrium (in pure strategies) of the game (2.6): H i ( x , . . . , x i , . . . , x N ) ≥ H i ( x , . . . , x i , . . . , x N ) , j = 1 , . . . , N, x i ≥ . The existence and uniqueness of a Nash equilibrium (for N ≥
2) was proved in [30]. Theproof was simplified in [9], see also [10] for an exposition.It is easy to see that x has at least two positive components. Furthermore, for such x the functions x i H i ( x , . . . , x i , . . . , x N ) are (strictly) concave. An elementary analysis ofthe correspondent one-dimensional problems shows that x is characterized by the followingrelations ϕ ′ i ( x i ) = M s − x i s , if ϕ ′ i (0) < Ms , (2.7) x i = 0 , if ϕ ′ i (0) ≥ Ms , (2.8) s = N X j =1 x j . (2.9)Following [30], note that for s > s ϕ ′ i ( z i ) = M ( s − z i ) , sϕ ′ i (0) < M, (2.10) z i ( s ) = 0 , sϕ ′ i (0) ≥ M uniquely define continuous functions z i ( s ). Clearly, x is a Nash equilibrium iff x = z ( s ),where s is a solution of the equation N X i =1 z i ( s ) = s, s > . (2.11) SYMPTOTIC EFFICIENCY OF THE PROPORTIONAL COMPENSATION SCHEME 5
Following [9, 10] let us pass from the replacement functions z i to the share functions σ i ( s ) = z i ( s ) /s . The existence and uniqueness of a Nash equilibrium follow from (2.11) inview of the properties of the share functions (see [10, Proposition 2]): σ i are continuous,strictly decreasing where positive,lim s → σ i ( s ) = 1 , lim s →∞ σ i ( s ) = 0 . So, the equation N X i =1 σ i ( s ) = 1 , s > , (2.12)which is equivalent to (2.11), has a unique solution. Remark 1.
Introducing the game (2.6), we followed the reasoning of [19]. In their model(inspired by [20]), the users share a communication link of some given capacity. The linkmanager gets payments (bids) from the users and allocates the rates according to the an-nounced price. The manager adjusts the price in order to allocate the entire link capacity.If the users are price takers, then the model is referred to as a competitive equilibrium. Ifthey are price-anticipating, then they are involved in a game, and it is assumed that theirbids correspond to a Nash equilibrium.The reward functions (2.3), in fact, only tell the agents the production levels x ∗ i − ε i ,specified for them by the manager. This normative scheme is quite sensible to individualcost functions ϕ i . Moreover, it does not announce any common compensation rules. All thesedrawbacks force to seek for more reliable compensation schemes. The following discussionshows that this task is not trivial.We see that (2.2) is a solvable convex optimization problem, satisfying the Slater condition.Hence, x ∗ ≥ λ ∗ ≥ λ ∗ ϕ ′ j ( x ∗ j ) = 1 , if x ∗ j > λ ∗ ϕ ′ j (0) ≥ , if x ∗ j = 0; (2.13) λ ∗ N X i =1 ϕ i ( x ∗ i ) − M ! = 0 , N X i =1 ϕ i ( x ∗ i ) ≤ M. Equivalently, x ∗ ≥ λ ∗ > N X i =1 ϕ i ( x ∗ i ) = M (2.14)hold true. Furthermore, for given λ ∗ > x ∗ ≥ x ∗ i is anoptimal solution of the problem x i /λ ∗ − ϕ i ( x i ) → max x i ≥ (2.15)similar to (2.4). Assume for a moment that ϕ i are strictly convex. Then (2.14) implies that x ∗ is unique and λ ∗ is also uniquely defined by (2.13), since at least one component of x ∗ ispositive.It is tempting to try ψ i ( x i ) = x i /λ ∗ for the role of reward functions. Indeed, by (2.15),they stimulate optimal production levels x ∗ i . However, in contrast to the piece rate scheme, DMITRY B. ROKHLIN AND ANATOLY USOV ψ i ( x i ) = x i /λ ∗ are not legal reward functions, since x ∗ i /λ ∗ − ϕ i ( x ∗ i ) > x ∗ i >
0, and thetotal reward exceeds the budget M : N X i =1 ψ i ( x ∗ i ) = 1 λ ∗ N X i =1 x ∗ i > N X i =1 ϕ i ( x ∗ i ) = M. Note, that the substitution x i = ϕ − i ( y i ) reduces (2.2) to the following equivalent problem:sup ( N X i =1 U i ( y i ) : N X i =1 y i ≤ M, y ≥ ) , (2.16)where U i ( y i ) = ϕ − i ( y i ) are strictly increasing concave functions. This is a customary non-linear resource allocation problem: see, e.g., [28]. If the functions U i are strictly concave,then, similarly to the above discussion, there is a unique pair ( y ∗ , µ ∗ ) with y ∗ ≥ µ ∗ > U ′ i ( y ∗ i ) = µ ∗ , if y ∗ i > U ′ i (0) ≤ µ ∗ , if y ∗ i = 0; N X i =1 y ∗ i = M. It follows that the unique optimal solution y ∗ of (2.16) can be recovered from the one-dimensional optimization problems U i ( y i ) − µ ∗ y i → max y i ≥ . (2.17)Thus, by selling the resource at price µ ∗ (per unit), the manager can stimulate the optimalplan y ∗ . But in the present context y i = ϕ i ( x i ) correspond to production costs, so theoptimization problems (2.17) make no economic sense. Remark 2.
Closing this section, we will show that any contest scheme cannot producebetter result than (2.2). Consider a non-cooperative game between N agents with the payofffunctions H i ( x ) = Ψ i ( x , . . . , x N ) − ϕ i ( x ) , x ≥ , where Ψ i ≥ i -th agent, and Ψ i (0) = 0. Let a random vector ( ξ , . . . , ξ N ) ≥ E (Ψ i ( ξ , . . . , ξ n ) − ϕ i ( ξ i )) ≥ E (Ψ i ( ξ , . . . , ξ i , . . . , ξ n ) − ϕ i ( ξ i )) , ξ i ≥ . We implicitly assume that all expectations exist. Putting ξ i = 0, we infer that E (Ψ i ( ξ , . . . , ξ n ) − ϕ i ( ξ i )) ≥ . If the total reward on average does not exceed M : P Ni =1 E Ψ i ( ξ , . . . , ξ n ) ≤ M , then N X i =1 E ϕ i ( ξ i ) ≤ M. A fortiori, P Ni =1 ϕ i ( E ξ i ) ≤ M by the Jensen inequality, and from the definition (2.2) of s ∗ itfollows that N X i =1 E ξ i ≤ s ∗ . SYMPTOTIC EFFICIENCY OF THE PROPORTIONAL COMPENSATION SCHEME 7
This negative result is by no means an indication that contest compensation schemes areuseless. The point is that the organization of a contest may not require the knowledge ofproduction cost functions ϕ i .3. The prices of anarchy in case of a large number of producers
In this paper we are interested in the behaviour of the following two versions of the “priceof anarchy”: A N = s ∗ /s, A ′ N = ˆ s/s for a large number N of agents. Recall, that the quantities s ∗ , ˆ s , s describe three types ofthe aggregate production:(i) s ∗ corresponds to an optimal cooperative strategy, enforced by the reward functions(2.3): see (2.2) (the normative compensation scheme);(ii) ˆ s is related to the case of “reward-taking” agents: see (2.4), (2.5), where the “best”common linear reward function is announced by the manager (the linear piece ratescompensation scheme);(iii) s corresponds to the Nash equilibrium of the game (2.6) for “reward-anticipating”agents (the proportional compensation scheme).We will refer to the related problems as (i), (ii) and (iii).The case of identical cost functions is considered in the following theorem. Theorem 1.
Assume that ϕ i = ϕ and ϕ ( y ) ∼ cy α , ϕ ′ ( y ) ∼ αcy α − , y → +0; c > , α ≥ . (3.1) Then lim N →∞ A N = α /α , lim N →∞ A ′ N = 1 . Proof. (i) As it was mentioned in Section 2, a vector x ∗ is an optimal solution of (2.2) iff itsatisfies (2.14), and there exists λ ∗ ≥
0, satisfying (2.13). It is natural to seek a solution ofthe (2.13), (2.14) in the symmetric form: x ∗ i = y ∗ > i = 1 , . . . , N . We have λ ∗ ϕ ′ ( y ∗ ) = 1 , λ ∗ ≥ N ϕ ( y ∗ ) = M. (3.2)Clearly, such a pair ( y ∗ , λ ∗ ) exists. From the second equality (3.2) it follows that y ∗ → N → ∞ , and using the first condition (3.1), we get y ∗ ∼ (cid:18) McN (cid:19) /α , s ∗ = N y ∗ ∼ (cid:18) Mc (cid:19) /α N ( α − /α , N → ∞ . (3.3)(ii) From (2.4), (2.5) we see that ˆ x i are identical: ˆ x i ( µ ) = ˆ y , and ϕ ′ (ˆ y ) = µ, N µ ˆ y = M. (3.4)Using (3.4) and (3.1), we conclude that ˆ y → N → ∞ and µ ˆ y = MN = ˆ yϕ ′ (ˆ y ) ∼ αc ˆ y α ; ˆ y ∼ (cid:18) MαcN (cid:19) /α , ˆ s = N ˆ y ∼ (cid:18) Mαc (cid:19) /α N ( α − /α , N → ∞ . (3.5) DMITRY B. ROKHLIN AND ANATOLY USOV (iii) We look for a symmetric Nash equilibrium of (2.6): x i = y > i = 1 , . . . , N . From(2.7), (2.9) we get ϕ ′ ( y ) = M N − N y . Hence, y → N → ∞ and M N − N = yϕ ′ ( y ) ∼ αcy α ; y ∼ (cid:18) MαcN (cid:19) /α ,s = N y ∼ (cid:18) Mcα (cid:19) /α N ( α − /α , N → ∞ . (3.6)The assertion of the theorem follows from the asymptotic forms (3.3), (3.5), (3.6). (cid:3) Assume that the agents use the same technology, but obtain resources at different prices.This situation is natural if the firm has departments in various locations. In this case theresource prices may depend on the quality of transportation network, the cost of labour,etc., in a concrete location. Denote by ( r , . . . , r m ) the resource amounts (inputs), andby ( p i , . . . , p im ) their prices in i -th location. For the production function F ( r , . . . , r m ) theproduction cost function is defined by ϕ i ( x ) = inf ( m X j =1 p ij r j : F ( r , . . . , r m ) ≥ x, r ≥ ) . For the Cobb-Douglas production function F ( r ) = A Q mj =1 r β j j , A > β j > ϕ i ( x ) = inf ( m X j =1 p ij r j : m X j =1 β j ln r j ≥ ln( x/A ) , r ≥ ) = sup λ ≥ θ i ( λ ) , where ln 0 = −∞ and θ i ( λ ) = inf r ≥ ( m X j =1 p ij r j + λ ln( x/A ) − m X j =1 β j ln r j !) = λ ln( x/A ) + m X j =1 inf r j ≥ { p ij r j − λβ j ln r j } = λ ln( x/A ) + m X j =1 (cid:18) λβ j − λβ j ln λβ j p ij (cid:19) . An elementary calculation shows that ϕ i ( x ) = sup λ ≥ θ i ( λ ) = c i x α , α = 1 P mj =1 β j , c i = 1 αA α m Y j =1 (cid:18) p ij β j (cid:19) β j ! α . Similarly, for the generalized CES production function (see, e.g., [4, 31]): F ( r ) = A m X j =1 a ρj r ρj ! γ/ρ , A, a j , γ > , ρ ∈ (0 , SYMPTOTIC EFFICIENCY OF THE PROPORTIONAL COMPENSATION SCHEME 9 we have ϕ i ( x ) = inf ( m X j =1 p ij r j : m X j =1 a ρj r ρj ≥ (cid:16) xA (cid:17) ρ/γ , r ≥ ) = sup λ ≥ θ i ( λ ) , where θ i ( λ ) = inf r ≥ ( m X j =1 p ij r j + λ (cid:16) xA (cid:17) ρ/γ − m X j =1 a ρj r ρj !) = λ (cid:16) xA (cid:17) ρ/γ + m X j =1 inf r j ≥ { p ij r j − λa ρj r ρj } = λ (cid:16) xA (cid:17) ρ/γ − (1 − ρ ) m X j =1 (cid:18) a j ρp ij (cid:19) ρ/ (1 − ρ ) λ / (1 − ρ ) . Maximizing this expression over λ ≥
0, we get ϕ i ( x ) = c i x /γ , c i = 1 A /γ m X j =1 (cid:18) a j p ij (cid:19) ρ/ (1 − ρ ) ! − (1 − ρ ) /ρ . Thus, the Cobb-Douglas and generalized CES production functions with decreasing re-turns to scale (that is, with P mj =1 β j ≤ γ ≤ ϕ i ( x ) = c i x α , α ≥ . Certainly, this fact is known (see [11, Chapter 5]), and weonly recalled it here.Now we have enough economic motivation to consider a model, representing heterogeneousagents by power cost functions with common exponent and different multiplication constants.
Theorem 2.
Assume that ϕ i ( x ) = c i x α , α > , c i > and lim N →∞ N X i =1 (cid:18) min ≤ k ≤ N c k c i (cid:19) / ( α − = ∞ . (3.7) Then lim N →∞ A N = α /α , lim N →∞ A ′ N = 1 . (3.8) Proof. (i) By the Lagrange duality the value of the problem (2.2) can be represented asfollows: s ∗ = − inf ( − N X i =1 x i : N X i =1 c i x αi ≤ M, x ≥ ) = − sup λ ≥ θ ( λ ) ,θ ( λ ) = inf x ≥ ( − N X i =1 x i + λ N X i =1 c i x αi − M !) = − M λ + N X i =1 inf x i ≥ ( − x i + λc i x αi )= − M λ − Bλ − α − , B = ( α − (cid:18) α (cid:19) αα − N X i =1 (cid:18) c i (cid:19) α − . Maximizing this expression over λ ≥
0, we get s ∗ = M /α N X i =1 c / ( α − i ! ( α − /α . (3.9) (ii) The optimization problems (2.4) take the form µx i − c i x αi → max x i ≥ . Substituting their optimal solutions ˆ x i = ( µ/ ( c i α )) / ( α − in (2.5), we get µ = α /α M P Ni =1 c − / ( α − i ! ( α − /α . Hence, ˆ s = N X i =1 ˆ x i = (cid:16) µα (cid:17) / ( α − N X i =1 c / ( α − i = (cid:18) Mα (cid:19) /α N X j =1 c / ( α − j ! ( α − /α (3.10)Comparing with (3.9), we see that ˆ s = s ∗ /α /α . (iii) To analyse the proportional compensation scheme consider the equations (2.10),(2.11): χ i ( s, z i ) = s αc i z α − i − M s + M z i = 0 , (3.11) χ ( s, z ) = N X i =1 z i − s = 0 . For ˆ z i ( s ) = (cid:18) Mαc i s (cid:19) / ( α − , K ∈ (0 ,
1) (3.12)we have χ i ( s, ˆ z i ( s )) = M ˆ z i ( s ) > χ i ( s, K ˆ z i ( s )) = M ( K α − − s + M K ˆ z i ( s ) < , for s > s i ,s i = (cid:18) K − K α − (cid:19) ( α − /α (cid:18) Mαc i (cid:19) /α . A function χ i is strictly increasing in z i . Hence, the solution z i ( s ) of (3.11) satisfies theinequalities K ˆ z i ( s ) < z i ( s ) < ˆ z i ( s ) , s > s i . (3.13)Put g ( s ) = K N X i =1 ˆ z i ( s ) − s, χ ( s ) = N X i =1 z i ( s ) − s, h ( s ) = N X i =1 ˆ z i ( s ) − s. From (3.13) we get g ( s ) < χ ( s ) < h ( s ) , s > max ≤ i ≤ N s i . SYMPTOTIC EFFICIENCY OF THE PROPORTIONAL COMPENSATION SCHEME 11
For, ˆ s given by (3.10), we have h (ˆ s ) = N X i =1 (cid:18) Mαc i ˆ s (cid:19) / ( α − − ˆ s = 1ˆ s / ( α − (cid:18) Mα (cid:19) / ( α − N X i =1 c / ( α − i − ˆ s α/ ( α − ! = 0; g ( K ( α − /α ˆ s ) = K N X i =1 ˆ z i ( K ( α − /α ˆ s ) − K ( α − /α ˆ s = K ( α − /α N X i =1 ˆ z i (ˆ s ) − ˆ s ! = 0 . It follows that χ ( K ( α − /α ˆ s ) > , χ (ˆ s ) < , (3.14)if K ( α − /α ˆ s > max ≤ i ≤ N s i . The last inequality reduces to N X j =1 (cid:18) min ≤ i ≤ N c i c j (cid:19) / ( α − ≥ − K α − . (3.15)For any K ∈ (0 , N large enough.From (3.14) it follows that the unique solution s of (2.11), or, equivalently, of (2.12),satisfies the inequalities K ( α − /α ˆ s < s < ˆ s (3.16)for sufficiently large N , as far as χ ( s ) /s = P Ni =1 σ i ( s ) −
1, and the share functions σ i arestrictly decreasing. Since K ∈ (0 ,
1) is arbitrary, we conclude that s ∼ ˆ s, N → ∞ . (3.17)The assertion of the theorem follows from the relations (3.9), (3.10), (3.17). (cid:3) The limits (3.8) are similar to those of Theorem 1. Note thatlim N →∞ A N = α /α ≤ e /e ≈ . , α > . So, the asymptotic efficiency loss of the proportional compensation scheme, compared to thenormative one, does not exceed 31%.
Remark 3.
In any compensation scheme under the assumption (3.7) the total productiontends to + ∞ as N → ∞ . The same is true in the setting of Theorem 1 for α >
1. This isa consequence of the fact that the agent expenditures are very small for small amounts ofoutput. In other words, they can start production almost for free. By hiring a large numberof such agents the manager can ensure an arbitrary large output.
Remark 4.
The total cost to total premium ratio D N = 1 M N X i =1 c i z αi ( s ) can be regarded as a measure of the reward dissipation of the proportional compensationscheme. For the normative scheme this ratio always equals to 1: see (2.14). The estimates(3.13), (3.16) imply that K ˆ z i (ˆ s ) < K ˆ z i ( s ) < z i ( s ) < ˆ z i ( s ) < ˆ z i ( K ( α − /α ˆ s ) ,K α N X i =1 c i ˆ z αi (ˆ s ) < N X i =1 c i z αi ( s ) < N X i =1 c i ˆ z αi ( K ( α − /α ˆ s )for sufficiently large N . After the substitution of (3.12), (3.10) we get Mα K α = K α N X i =1 c i (cid:18) Mαc i ˆ s (cid:19) α/ ( α − < N X i =1 c i z αi ( s ) < K N X i =1 c i (cid:18) Mαc i ˆ s (cid:19) α/ ( α − = MαK .
Since K ∈ (0 ,
1) is arbitrary, it follows thatlim N →∞ D N = 1 /α > . Thus, the reward does not completely dissipate in the limit. This fact is also a consequenceof the inequality D N ≤ ( N − / ( N α ), obtained in [9, Theorem 5].The condition (3.7) is satisfied if the sequence c i is non-decreasing and ∞ X i =1 c / ( α − i = + ∞ . It is not satisfied for c / ( α − i = cq i − , c > q > q = 1. Remark 5.
Assume that c i are independent identically distributed (i.i.d.) random variables,bounded from below by a positive constant: c i ≥ c >
0. Then the condition (3.7) is satisfiedalmost surely: N X i =1 (cid:18) min ≤ k ≤ N c k c i (cid:19) / ( α − ≥ c / ( α − N X i =1 (cid:18) c i (cid:19) / ( α − → ∞ a.s. , N → ∞ . Indeed, consider a sequence of strictly positive i.i.d. random variables ξ i and take a constant L > ν = E ( ξ i ∧ L ) >
0, where a ∧ b = min { a, b } . By the strong law of largenumbers we have lim inf N →∞ N N X i =1 ξ i ≥ lim N →∞ N N X i =1 ξ i ∧ L = ν a.s.Hence, P ∞ i =1 ξ i ≥ P ∞ i =1 ξ i ∧ L = + ∞ .The case of linear cost functions ϕ i ( x ) = c i x appears to be more complex from the asymp-totical point of view, although there is known an explicit expression for s in this case: see[17, Proposition 5]. For reader’s convenience, in the next theorem we derive this expressionusing the argumentation similar to [32, Theorem 4.19]. We only consider the price of anarchy A N , since from (2.4), (2.5) we see that µ = max ≤ i ≤ N c i , and in this case optimal productionlevels ˆ x i of reward-taking agents either equal to zero or are not uniquely defined. SYMPTOTIC EFFICIENCY OF THE PROPORTIONAL COMPENSATION SCHEME 13
Theorem 3.
Assume that ϕ i ( x ) = c i x , < c ≤ · · · ≤ c N . Then A N = 1 l − l X i =1 c i c , (3.18) l = min { i ∈ { , . . . , N − } : 1 i − i X k =1 c k ≤ c i +1 } ∧ N, where l is the number of active players (we use the convention min ∅ = + ∞ ). Moreover, c c < A N ≤ c c . (3.19) Proof.
From (2.2) by the Lagrange duality we get s ∗ = − inf ( − N X i =1 x i : N X i =1 c i x i ≤ M, x ≥ ) = − sup λ ≥ θ ( λ ) ,θ ( λ ) = inf x ≥ ( − N X i =1 x i + λ N X i =1 c i x i − M !) = − λM + N X i =1 inf x i ≥ ( λc i − x i = ( − λM, λ ≥ /c , −∞ , λ < /c . Thus, s ∗ = M/c . It also follows that x ∗ = M/c , x ∗ i = 0, i ≥ c .Putting σ i = x i /s , rewrite the equations (2.7) – (2.9), determining the equilibrium of theproportional compensation scheme, as follows: Mc i (1 − σ i ) = s, s < Mc i , (3.20) x i = 0 , s ≥ Mc i , N X i =1 σ i = 1 , σ i ≥ , s > . (3.21)Denote by l the number of active players. Then l = max { i : s < M/c i } , s < M/c l , s ≥ M/c l +1 . (3.22)Using the equality (3.21), from (3.20) we get s = M l − P li =1 c i . (3.23)As we know, there are at least two active players. Thus, from (3.22), (3.23) we concludethat l can be expressed as follows l = min ( i ∈ { , . . . , N − } : i − P ik =1 c k ≥ c i +1 ) ∧ N. This formula gives (3.18): A N = s ∗ s = 1 l − l X i =1 c i c . The inequalities of the form (3.19) are presented in [32, Corollaries 4.21, 4.22]. The leftinequality (3.19) is implied by (3.22): s < M/c l ≤ M/c . From (3.20) it follows that σ ≤ σ .Hence, σ ≤ / s = Mc (1 − σ ) ≥ M c . This gives the right inequality (3.19). (cid:3)
One more estimate A N ≤ N − N X i =1 c i c , N ≥ l = N this estimate turns into the equality. Furthermore, if2 ≤ l < N , then l − P lk =1 c k ≤ c l +1 , and (3.24) is implied by the inequality1 N − N X k =1 c k = 1 N − l X k =1 c k + 1 N − N X k = l +1 c k ≥ N − l X k =1 c k + N − lN − c l +1 ≥ N − l X k =1 c k + N − lN − l − l X k =1 c k = 1 l − l X k =1 c k . To obtain a meaningful asymptotic result, let us assume, as in Remark 5, that c i are i.i.d.random variables and c i ≥ c >
0. Then, by the strong law of large numbers, from (3.24) itfollows that lim N →∞ A N ≤ νc a.s. , where ν = E c i . However, our numerical experiments suggest a much better result. Conjecture 1.
Assume that ϕ i ( x ) = c i x , where c i are i.i.d. random variables such that c i ≥ c >
0. Then lim N →∞ A N = 1 a.s.Denote by c ( N ) k the k -th order statistics of the sequence ( c i ) Ni =1 . That is, c ( N ) k is a k -thsmallest element of the sequence ( c i ) Ni =1 : c ( N )1 ≤ · · · ≤ c ( N ) N , c ( N )1 = min { c , . . . , c N } , . . . , c ( N ) N = max { c , . . . , c N } . From (3.22) it follows that A N = s ∗ s ≤ c ( N ) l +1 c ( N )1 . To prove the Conjecture 1 it would be enough to show that c ( N ) l +1 /c N → l = min ( i ∈ { , . . . , N − } : 1 i − i X k =1 c ( N ) k ≤ c ( N ) i +1 ) ∧ N. But this is not an easy task.
SYMPTOTIC EFFICIENCY OF THE PROPORTIONAL COMPENSATION SCHEME 15
Table 1.
Sampled values of A N − c i = 1 + ξ i , where the distributionsi.i.d. random variables ξ i are indicated in the first row. N U (1 , U (1 , LN (0 , LN (0 ,
2) Pa(0 . ,
1) Pa(3 , . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − Table 2.
Sampled values of the proportion l/N of active players for c i = 1+ ξ i ,where the distributions i.i.d. random variables ξ i are indicated in the first row. N U (1 , U (1 , LN (0 , LN (0 ,
2) Pa(0 . ,
1) Pa(3 , . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − The numerical experiments (implemented by the means of the R software), supporting thisconjecture, are presented in Table 1. As usual, by U ( a, b ) we denote the uniform distributionon ( a, b ). We write ξ ∼ LN ( µ, σ ), if ξ = e µ + ση , where η is a standard normal randomvariable. The density of the Pareto distribution Pa( α, λ ) is given by the formula f ( x ) = αλ α ( λ + x ) α +1 , x > . Note that neither a heavy tail nor the infinite expectation (for α = 0 .
5) of the Paretodistribution prevent the convergence A N → N → ∞ .Table 2 indicates that under the assumptions of Conjecture 1 the proportion of activeplayers l/N tends to zero. Remark 6.
Concluding the paper, we recall a popular concept of no-regret learning , ex-plaining the emergence of an equilibrium in a repeated game. Consider a game with thepayoff functions u i ( x , . . . , x N ), i = 1 , . . . , N , defined on S × · · · × S N , and assume that anagent i knows only his own payoff u i and picks his strategy x ti ∈ S i according to a no-regretalgorithm , ensuring thatmax y ∈ S i T X t =1 u i ( y, x t − i ) − T X t =1 u i ( x ti , x t − i ) = o ( T ) , T → ∞ , (3.25) where x t − i = ( x tk ) k = i . Under appropriate assumptions, a plenty of such algorithms is providedby the theory of online convex optimization: see, e.g., [16].In [13] for the class of socially concave games it was proved that the average strategy vector T P Tt =1 x t converges to a Nash equilibrium, if x t satisfies the no-regret property (3.25). 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