Centralization or Decentralization in Multi-Agency Contracting. Games?

Centralization or Decentralization in Multi-Agency Contracting Games? Yu Chen 19th February 2014 Abstract We explore the connection and comparison between the decentralized menu contracting procedure and the centralized mechanism contracting procedure in generalized multi-agency games with "-ex post implementation. We identify that "-ex post menu design is only strategically equivalent to individual-based "-ex post mechanism design rather than jointbased "-ex post mechanism design. We thus present a comparative analysis. We show that joint-based "-ex post mechanism design always weakly dominates individual-based "-ex post mechanism design and "-ex post menu design. We also provide economically interesting conditions for when the former strictly dominates the latter two and when they are all equivalent. Keywords: multi-agency, "-ex post equilibrium, mechanism design, menu design, delegation principle JEL Classi cation: C79 D82 D86 his paper is a revision of Chapter I of my Ph.D. dissertation at Indiana University. I would like to thank Frank Page and Robert Becker for advice, encouragement, and comments. For additional helpful suggestions and comments, thanks are also due to Seungjin Han, Kim-Sau Chung, Yongchao hang, Ronald M. Giammarino, Michael Koss, Sascha Baghestanian, Filomena Garcia, Haomiao Yu, Ran Shao, Phil Reny and the participants/audiences at Shanghai Jiaotong University, hejiang University, Nanjing University, PE annual meeting 2012, Indiana University Microeconomics Workshop Fall 2012, and St Louis Fed/Missouri Economics Conference 2012. I am solely responsible for any errors. Please address correspondence to: Yu Chen, School of Economics, Nanjing University, 16 Jinyin Street, Anzhong building Rm 2410, Nanjing, Jiangsu 210009, China; (+86)25-83621970, yourchenyu@gmail.com. 1

1 1 Introduction his paper explores the connection and comparison between the decentralized menu contracting procedure and the centralized mechanism contracting procedure in generalized multi-agency contracting games with adverse selection. 1 Here multi-agency denotes that a single principal (in short, PL) contracts with multiple agents. Multi-agency situations are more realistic and more frequently observed than single-agency situations in principal-agent relationships. he regulator needs to regulate oligopolists in the market. he government will provide public goods and impose tax on individuals. A company owner needs to compensate the employees. Compared with the single-agency environment, the generalized multi-agency environment implies that the impacts of di erent agents asymmetric information on PL s objective may be interrelated. First, PL s payo can jointly depend on all the agents types and contract speci cations. Second, the agents are allowed to be not only heterogeneous but also "fully interdependent." Each agent can have a heterogeneous payo depending on not only his own private type and contract speci cation but also those of other agents. Moreover, correlated types or some feasible constraints on joint contract selection are permitted across the agents. Accordingly, the agents will strategically interact with each other. Such interrelated impacts and strategic behaviors of the agents turn out to be pervasive in modern society. o resolve the adverse selection problems, PL needs to nd a way to link a pro le of contracts for the agents with each pro le of their private information, so as to create incentives for the agents to enter into a contract-selection pro le in her 2 best interest. 3 Hence, we can identify two major classes of contracting procedures to implement this goal, considering whether specifying contracts for the agents, as a key decision in contracting games, can be centralized or decentralized. One class is centralized mechanism design. It requires the agents to report some messages to PL after PL proposes a communication mechanism associating contract-pair pro- les with report pro les. PL will then specify the contract for each agent after receiving their reports. PL needs to design an optimal mechanism in this case. he other class is decentral- 1 Although this paper focuses on pure adverse selection, it is not technically demanding to include moral hazard in this context. 2 hroughout the paper, masculine pronouns refer to the agents, whereas feminine pronouns apply to the principal. 3 Adverse selection is actually regarded as the primary concern.

2 ized menu 4 design. PL proposes to the agents a joint menu, namely, a set of contract pro les. he agents are entitled to directly choose a contract within the pre-o ered menu and an in the known set. his implies decentralization of the decision rights to specify contracts. PL needs to design an optimal menu in this case. Menu design is more straightforward and can skip explicit information communication in specifying contracts. Many authors study the menu design as a di erent self-contained contracting procedure from mechanism design, 5 since these two procedures apparently have distinct economic meanings. People have been interested in whether contracting is decentralized, in other words, whether centralized mechanisms are decentralizable by menus, that is, whether decentralized menu design are strategically equivalent to centralized mechanism design. he meaning of strategic equivalence between mechanism design and menu design is as follows: if there is an optimal mechanism solving the mechanism design problem, there also exists an optimal menu solving the menu design problem, vice versa, and solutions to these two problems brings the same (expected) payo to PL. Previous studies demonstrate that such strategic equivalence can always be achieved in the single-agency situations. his fact is called the delegation principle. Page (1992) and Carlier (2001) characterize the incentive compatible direct mechanisms for single-principal single-agent contracting problems via the sets of contracts. Peters (2001), Martimort and Stole (2002), and Page and Monteiro (2003) study the strategic equivalence of menu design and mechanism design in multi-principal single-agent contracting problems. However, the interrelated impacts of the agents asymmetric information and strategic behaviors among the agents bring to generalized multi-agency contracting problems signi cant structural complications. So it is not straightforward to nd a basis of the connection or comparison between centralization and decentralization immediately by extending the delegation principle for single-agency. here have not been any rigorous studies on whether these two self-contained contracting problems, menu design problem and mechanism design problem, are strategically equivalent in generalized multi-agency contracting environments. If they are not strategically equivalent in general, it is also important to nd a way to know which one will be 4 Some authors use the term "catalogs" instead. 5 See Martimort and Stole (2002), Page(1992), Page and Monteiro (2003), Carmona and Fajardo(2007) among many others.

3 more desirable under what conditions and when the strategic equivalence can still be achieved. his paper addresses these questions. In generalized multi-agency contracting games, centralization of the right to specify contracts and corresponding communication between PL and the agents may be more desirable than decentralization in many cases. Although decentralized menu design is a "simpler" procedure, 6 it can only incorporate separate, individual information to deal with the interrelated information asymmetry problem. In contrast, centralization can incorporate more comprehensive information and use relative information evaluation. A loss of generality will be imposed if one still takes decentralization for granted in multi-agency. his paper makes three contributions to principal-agent theory. he rst contribution is a comprehensive formulation of relevant multi-agency contracting problems based on a generalized model setting. his describes how PL should structure contracting procedures to deal with the agents interrelated impact of asymmetric information and strategic behaviors. he multi-agency contracting game of our interest is essentially a one-shot pure-strategy two-stage game. PL s strategy is to o er mechanisms or joint menus for the agents. Each pre-o ered mechanism or joint menu actually de nes a non-cooperative subgame for all the agents to play simultaneously. In this paper, "-ex post equilibrium ("-EPE) is of our interest as the solution concept of such a subgame. It is a practical and robust solution concept. It re ects the bounded rationality of human beings and the approximation of exact equilibria, and it does not require each agent s probability assessment about the others private information to be common knowledge. ("- )EPE has been increasingly studied in game theory (especially see Kalai 2004). McLean and Postlewaite (2013) further studies the occasion on which truthful revelation is an "-EPE in mechanism design. Generally, in the mechanism design, PL seeks an optimal mechanism from a class of pre-o ered mechanisms inducing a subgame for the agents in which some particular participation strategy pro le of the agents will be achieved as "-EPE. Such class of mechanisms is called "-ex post mechanisms. In the menu design, PL seeks an optimal joint menu from a class of pre-o ered joint menus inducing a subgame for the agents in which some particular contract-selection strategy pro le of the agents will be achieved as "-EPE. Such class of joint menus is called "-ex post (joint) menus. In this respect, the contracting games over either 6 Precisely, decentralization may need lower transaction cost of using menus than that of using mechanisms.

4 "-ex post mechanisms or "-ex post menus are referred to as contracting games with "-ex post implementation. Moreover, our analysis allows certain feasible constraints to be imposed on the joint contract set, i.e. the set of contract pro les for all agents. his is consistent with many realistic situations, such as optimal auction design, budget constraint, etc. In that case, the subgames induced by mechanisms or menus are related to the concept of "generalized games." 7 he second contribution is the delegation principle for "-ex post implementation in this generalized multi-agency contracting environment. his principle identi es that "-ex post menu design are strategically equivalent to individual-based "-EPIC mechanism design rather than joint-based "-EPIC mechanism design. Here "joint-based" mechanisms associate the contracts for an individual agent with the joint reports of all agents, whereas "individual-based" mechanisms associate the contracts for an individual agent merely with his individual reports. One may regard the range of any individual-based "-EPIC mechanism as a joint menu. Such range is the image of the mechanism mapping over the message space. Yet it is not straightforward to nd that the range of optimal individual-based "-EPIC mechanism must be an optimal joint menu in the generalized multi-agency problem over menus or bring identical welfare to PL. It is also not trivial to see that an optimal "-ex post joint menu can de nitely induce an optimal individualbased "-EPIC mechanism in the generalized multi-agency problem over mechanisms or bring identical welfare to PL. A rigorous proof of the delegation principle is provided by construction under a general mathematical setting allowing in nite/continuum structures. Nevertheless, individual-based mechanisms are still realistic in several situations. According to Dequiedt and Martimort (2010), individual-based mechanisms are still adopted in practice due to some legal customs, technology restrictions, etc. he last but most important contribution is a comparative analysis between di erent contracting procedures. By the delegation principle for "-ex post implementation, individual-based "-EPIC mechanisms can serve as an important bridge for the comparison between centralization and decentralization in multi-agency from PL s perspective. he set of individual-based 7 Generalized games are also called constrained games, shared constraint games, games with coupled constraints, or abstract economies. he systematic description of "generalized games" in the normal form with complete information appears in the work of Rosen (1965) and Ponstein (1966). Since EPE is the strategy pro le under which every action pro le is just a Nash equilibrium at every type pro le, the existing studies on Nash equilibria of "generalized games" can still be connected to this context.

5 "-EPIC mechanisms is actually equivalent to a subset of joint-based "-EPIC mechanisms. We also characterize these two in view of "weak monotonicity." Clearly, joint-based "-EPIC mechanism design always makes PL better o than individual-based "-EPIC mechanism design and "-ex post menu design. But this is just a weak dominance result. We also present several sets of economically interesting conditions for when joint-based "-EPIC mechanism design can strictly dominate individual-based "-EPIC mechanism design and "-ex post menu design. We also characterize the joint-based and individual-based EPIC mechanisms to compare them in classic quasilinear environments. Meanwhile, we show that all the relevant contracting procedures must be equivalent when the impacts of di erent agents asymmetric information on PL s objective are essentially independent and separate. 8 Our results are of more signi cance if there are economically interesting environments where the existence of nontrivial "-ex post implementation, namely, nonconstant (or nontrivial) optimal "-EPIC mechanism or non-singleton optimal "-ex post menu, can be ensured. 9 Exact EPE or nontrivial ex post implementation in the "full-blown" form has been criticized for failing to exist under mild conditions. 10 he issue of existence would usually be alleviated in the context of "-EPE or "-ex post implementation. In addition, our results will still hold when there is a degenerated form of "full interdependence" among the agents, that is, when each agent s payo only depends either on his own type i.e., private valuation case in which "-ex post implementation will also degenerate to "-dominant strategy implementation, or on his own contract, i.e., no contract externalities. In the degenerated form, "-EPE or "-ex post implementation is normally more likely to exist. Hammond (1979) and Han (2006) also address the issue of decentralization in multi-agency games with pure adverse selection, private valuations, and independent feasible contract sets of individual agents. 11 Hammond suggests that a dominant strategy incentive compatible mechanism is decentralizable by a pro le of some particular contract sets indexed by type pro les. Han 8 here is a relatively trivial occasion for this as well. If the only optimal joint-based "-EPIC mechanisms are constant mechanisms, the optimal individual-based mechanisms will take the same constants. 9 By the delegation principle for "-ex post implementation, if the optimal individual-based "-EPIC mechanism is constant, then there exists a singleton optimal "-ex post menu, vice versa. Yet, the optimal joint-based "-EPIC mechanism is not necessarily constant in this situation. If it is constant, then the optimal individual-based "-EPIC mechanism must be constant, and the optimal "-ex post menu must be singleton. 10 A well-known counterexample is raised by Jehiel, Moldovanu, Meyer-er-Vehn, and ame (2006). 11 It implies the feasible contract sets of individual agents are not cross-constrained.

6 constructs a multi-principal multi-agent "bilateral contracting" environment restricting feasible mechanisms. All "bilateral" mechanisms the principals can o er to each agent are just the functions from a single uniform message set to an independent set of contracts available to that agent. He shows that "bilateral" mechanism design is strategically equivalent to menu design under Perfect Bayesian Equilibrium in "bilateral contracting" games. Instead, in this paper, (1) we extend the models of Hammond and Han to a much more general situation in multi-agency games. he agents can be fully interdependent, and feasible contract sets of individual agents can be heterogeneous and cross-constrained. he message sets of di erent agents can be heterogeneous or unidentical. he mechanisms are allowed to be joint-based; (2) decentralization or delegation in this paper is centered on self-contained menu design games in which menus are the sets of contracts independent of any index. his is more realistic in economic practices; (3) "-ex post implementation, instead of Bayesian implementation or dominant strategy implementation, is addressed in this paper. Moreover, Dequiedt and Martimort s work (2010) also extends Han s setting to allow message sets of di erent agents to be heterogeneous or unidentical in the single-principal multi-agent bilateral contracting environment with private valuations and Bayesian implementation. he authors claim the rationale of using individual-based mechanisms in some real-life situations and treat the ranges of individual-based mechanisms over the message sets as menus. Nonetheless, the authors focus on examining how PL acts opportunistically in the environment. hey have not substantively discussed the strategic equivalence or dominance between mechanism design and menu design in a generalized multi-agency contracting environment. he rest of the paper is organized as follows. Section 2 presents basic elements of the model and provides some discussion and several examples within the model s scope. Section 3 explores the "-ex post mechanism design problem and proves the related revelation principle. Section 4 investigates the menu design problem and shows the delegation principle for "-ex post implementation. he comparative analysis between centralization and decentralization is further presented in Section 5. Section 6 applies the main results to incentive-based nancial regulation with interrelated banks and discusses why the regulatory contracting cannot be decentralized without loss of generality. Concluding remarks are given in section 7.

7 2 Preliminaries here are one PL and n agents indexed by i 2 N = f1; ; ng in the multi-agency contracting game. PL moves rst, and then the agents follow simultaneously and behave non-cooperatively. hroughout this paper, the symbol B(X) is reserved for Borel -algebra of a certain space X, and the symbol M(X; Y ) is reserved for the set of all (B(X); B(Y ))-measurable functions from one space X to another space Y. 2.1 Agent ypes Each agent i (in short, A i ) has private payo type that is only observable to himself. It is denoted by t i 2 i, where i is a Borel space. 12 Q We write t = (t i ) i2n 2 = n i and Q t i = (t i ) i2n nfjg 2 i = n j. Let i be a probability measure de ned on B( i ) and be a j6=i probability measure on the associated product Borel -algebra B( ). hus, and i are also Borel spaces. (; B( ); ) is a probability measure space characterizing the common prior over the agents types. If the types of di erent agents are independent, = 1 n, i.e., is the product probability measure on B( ). Correlated types are also permitted. 2.2 Contracts All possible contracts available to A i is k i 2 K i. K i contains an element k 0 which denotes "no Q contracting". he feasible joint contract set is K n K i. Its element is k = (k i ) i2n 2 K. K may contain some realistic constraints on the contract pro les PL can o er to the agents. K is assumed to be a compact metric space. Many examples t these assumptions on (joint) contract sets in multi-agency situations. Example 1 (Joint Contract Sets). (1) Finite contract sets: here are only nitely many contracts in each K i. K is a compact nq metric space as any subset of K i. (2) Product-price pairs with budget constraint: Each buyer i is o ered a product-price pair (x i ; p i ): x i is some product characteristics, such as quantity, quality, etc. p i is the price the seller 12 i.e. a Borel subset of a Polish space. A complete separable metric space is called Polish space.

8 can charge for i. his setting is frequently observed in nonlinear pricing games. So the feasible joint contract set can be either K = f(x 1 ; ; x n ; p 1 ; ; p n ) 2 R n R n : 0 X i p i x i W g; where W > 0 is the budget upper bound of the seller, or K = f(x 1 ; ; x n ; p 1 ; ; p n ) 2 R n R n : 0 p i x i I i ; for each ig; where I i > 0 is the budget upper bound of buyer i. In either case, K is a compact metric space. (3) Contract sets for a single object: Each bidder i is o ered a pair (x i ; p i ): x i is i s payment to the seller. p i is the probability that i gets the object. his setting is frequently observed in optimal auction games. So the feasible joint contract set can be K = f(x 1 ; ; x n ; p 1 ; ; p n ) 2 R n R n : X i p i 1; p i 0; 0 x i I i ; for each ig; where I i > 0 is the wealth of bidder i. Obviously, K is a compact metric space. (4) Outcome-contingent contract sets: Assume that the contracts are outcome-contingent. 13 Q If for each i, (i) K i = M(; R), K = n Ki s, where each Ks i is a sequentially compact subset of K i for the topology of pointwise convergence on ;(ii) K s i contains no redundant contracts, that is, if for any two k i and k 0 i in Ks i satisfying k i (! 0 ) 6= k 0 i (!0 ) for some! 0 2, (f! 2 : k i (!) 6= k 0 i (!)g) > 0, and (iii) Ks i is uniformly bounded, then Ks i is compact and metrizable for the topology of pointwise convergence by Proposition 1 in ulcea (1973). So is K. 2.3 Payo s Let u : K! R denote PL s ex post payo function over contract and type pro les. Assume that u is continuous on K and Borel-measurable on. Moreover, u is -integrably bounded, i.e. there exists a -integrable function :! R such that for t almost everywhere with respect to, ju(k; t)j for all k 2 K. Let v i : K! R denote A i s ex post payo function de ned over contract pro les and type pro les. Assume that v i is continuous on K. 13 Page and Monterio (2003) raise a similar example for state-contingent contract sets in common agency.

9 his general payo setting permits the agents to be "fully interdependent." In detail, there are externalities in contracts among the agents because each agent s payo depends on not only his own contract but also the other agents contracts. Correlated types and interdependent valuations are permitted instead of private independent valuations, because each agent s outcome-expected payo is based on not only his own type but also the other agents types. In addition, PL s expected payo is permitted to depend jointly on all the agents types and contracts. Hence, the agents will behave strategically, and the impacts of their respective asymmetric information will be interrelated. Lastly, when making decisions the agents need to take into account the cost of discovering and using a better alternative strategy. It is denoted by " 2 [0; 1). It is common knowledge and related to bounded rationality of human beings. 2.4 Discussions he basic model setup is mainly based on general topological or metric structure and Borel measurability. It does not entail order structure or linear algebraic structure. Although some natural orders or vector spaces may exist in many applications, no monotonicity or concavity of payo s is assumed in the basic model setup. Di erentiability or validity of First Order Approach is not necessarily required as well. he following example address a few signi cant economic problems in real life. It satis es the assumptions proposed above. Example 2 (E cient mechanism design) Consider n agents in the standard social choice mechanism design problem. Each A i has his own private type t i 2 i = R: (t) denotes the common prior on the type pro les. Allocation y (x; s) 2 Y = (X; S); where x is the agents Q consumption choice and s is the agents transfers. Set X = n Q [x i ; x i ] and S = n [s i ; s i ]. P s i = 0:A i s quasilinear payo is v i (x; s; t) = h i (x; t i ) + s i. he "virtual" social planner s i P utility is u(x; s; t) = n h i (x; t i ).

10 3 Mechanism Design and Menu Design Problems 3.1 Contracting Game over "-Ex Post Mechanisms In the rst contracting procedure, centralized mechanism design, the principal-agent contracting game over mechanisms is played as below: At Stage 1, PL proposes a mechanism, which is commonly observable, to the agents. At Stage 2, each agent unilaterally learns his true type, and the agents simultaneously send reports to PL. At Stage 3, through the pre-o ered mechanism, PL assigns contracts to the agents according to their reports. At Stage 4, outcomes are realized and the contracts are implemented. Each mechanism o ered by PL induces a simultaneous-moved subgame of the agents with "-ex post equilibrium as the solution concept. Chung and Ely (2006) state a version of the revelation principle for ex post incentive compatibility together with ex post individual rationality in a pure adverse selection environment. heir result can be easily extended for the class of generalized pure strategy "-ex post mechanism games. So we can restrict our attention to the "-EPIC direct mechanisms out of the "-ex post general mechanisms with no loss of generality. Depending on the realistic situations, the feasible mechanisms can take two forms as below. De nition 1 Let k i :! K i denote a Borel measurable function specifying a contract to A i for each type report pro le of all agents. A "joint-based" direct mechanism is a pair of functions k 2F(; K), where k = (k 1 ; k n ) :! K satisfying (k 1 (t); ; k n (t)) 2 K for each t 2. De nition 2 Let k i : i! K i denote a Borel measurable function specifying a contract to A i for each type report of single A i. An "individual-based" direct mechanism is a pair of functions k 2 F(; K), where k = (k 1 ; k n ) : R! K satisfying (k 1 (t 1 ); ; k n (t n )) 2 K for each t 2. hus, each k in F(; K) and k in F(; K) are still Borel measurable. Each mechanism o ered by PL induces a simultaneous-moved subgame for the agents in which "-ex post equilibrium ("- EPE) is of interest as the solution concept. Several relevant de nitions are given as below.

11 De nition 3 A joint-based direct mechanism k is "-ex post incentive compatible ("-EPIC) if it induces truthful reporting as the "-EPE for all the agents, i.e. for each i 2 N and each t 2 ; v i (k(t); t) v i (k(t 0 i; t i ); t) "; (1) for all t 0 i 2 i. De nition 4 An individual-based direct contracting mechanism k is "-ex post incentive compatible if it induces truthful reporting as the "-EPE for all the agents, i.e., for each i 2 N and each t 2 ; v i (k(t); t) v i (k i (t 0 i); k i (t i ); t) "; (2) for all t 0 i 2 i. hroughout this paper, when the term ""-EPIC mechanisms" is used, it actually refers to the "-EPIC direct mechanism. In "-EPE, if a true type pro le is given, either the participation strategy pro le or truthful reporting and obedient acting is actually an "-Nash equilibrium. Also note that "-EPE is exact EPE if and only if " = 0. hus, two PL s optimization problems address contracting games over "-ex post mechanisms. (P1) joint-based "-EPIC mechanism design problem max u(k(t); t)(dt) k2f(;k) s.t. k is "-EPIC : (P1 0 ) individual-based "-EPIC mechanism design problem max u(k(t); t)(dt) k2f(;k) s.t. k is "-EPIC. Remark 1 PL may also consider maximizing her ex-post payo given any type pro le t instead.

12 But this will not bring structural change in the analysis of this paper. It is worth noting that the subgame induced by the pre-o ered mechanism may yield multiple equilibria. o solve the mechanism design problem, PL needs to select a particular equilibrium for tie-breaking. She may have su cient bargaining power to designate a particular equilibrium for the agents to play. In another case, she can recommend the agents to play a particular equilibrium. Due to the focal-point e ect, the agents will follow such a recommendation. 14 By choosing the optimal "-EPIC mechanism, the entire principal-agent contracting game can achieve an equilibrium. PL just focuses on truth-telling as the particular "-EPE under each direct mechanism. 3.2 Contracting Game over "-Ex Post Menus he other contracting procedure for PL to deal with asymmetric information in multi-agency is decentralized menu design. PL does not need to process decentralized information or have the agents send messages to specify contracts for the agents. Instead she can design a joint menu, i.e. a subset of the feasible joint contract set, for the agents. he agents are entitled to pick the contracts from the joint menu on their own accord. he possible contract menu for A i is C i K i. In view of feasible constraints in the joint contract set, the joint feasible menu is C 2 P f (K) f(c 1 ; ; C n )j(c 1 ; ; C n ) Kg, where P f (K) is a collection of nonempty, closed subsets of K. he principal-agent contracting game over menus is played as below: At Stage 1, PL proposes a joint contract menu, which is commonly observable, to the agents. At Stage 2, each agent unilaterally learns his true type. he agents simultaneously select the contracts from the pre-o ered joint menu. At Stage 3, outcomes are realized and the contracts are implemented. Each A i s strategy is a function e k i : i! K i. A i s contract selection according to his type is denoted by e k i 2 F i. Let e k = ( e k i ) i2n ; e k(t) = ( e k i (t i )) i2n ; e k i (t i ) = ( e k j (t j )) j2n nfig. Each menu o ered by PL induces a simultaneous-moved subgame played by the agents. Under a joint menu C, e k(t) 2 C for each t 2. his actually imposes a constraint on the strategy pro les of the 14 Myerson (1988) also talks about multiple equilibria and equilibrium selection in mechanism design in a survey.

13 agents. Let e k 2 F c = f e kj e k(t) 2 C for each t 2 g. Again, "-EPE is regarded as the solution concept of this constrained subgame. De nition 5 A contract selection pro le e k is called an "-ex post equilibrium under a joint menu C if for each i 2 N and each t 2, v i ( e k(t); t) v i ( e k i 0 (ti ); e k i (t i ); t) "; (3) for all e ki 0 2 F i satisfying ( e k i 0 (ti ); e k i (t i )) 2 C. Moreover, such a joint menu C is called an "-ex post (joint) menu. In "-EPE, if a true type pro le is given, the realized contract pair is an "-Nash equilibrium of the (constrained) subgame de ned by an "-ex post menu. PL can deduce that the agents will have the "-EPE contract selection pro le in the subgame. PL hence has an optimization problem to address this contracting game over "-ex post menus. (P2) "-Ex post menu design problem: R max u( e k(t); t)(dt) e k2fc max C2P f (K) s.t. e k is the "-EPE under C. Again, for tie-breaking in multiple equilibria PL may designate or recommend e k in her best interest for the agents with type pro le t to follow. hus, PL can link one contract pro le with each type pro le t. 4 he Delegation Principle for "-Ex Post Implementation We can completely characterize all "-ex post menus for contracting games using individualbased "-EPIC mechanisms indeed. his helps establish the strategic equivalence between the individual-based "-EPIC mechanism design problem and the "-ex post menu design problem, which is summarized by the delegation principle for "-ex post implementation. In this sense, all individual-based "-EPIC mechanisms can be decentralized via "-ex post menus. Consider the set-valued mapping " : P f (K) K de ned by " (t; C) = f e k(t) 2 C : e k is the "-EPE under Cg:

14 It is deducible by PL and represents the t-type-pro le agents joint "-ex post equilibrium response to any menu o er C. De nition 6 For any " 0 and C 2 P f (K), " (; C) is well-de ned if for each t, " (t; C) is nonempty. A well-de ned " (; C) suggests that the agents with each given type pro le t can possess at least one "-Nash equilibrium contract pair under the joint menu C. In other words, there exists at least one "-ex post menu C. he following proposition describes the complete characterization of all individual-based "-EPIC mechanisms via "-ex post menus. Proposition 1 For every " 0, given a contracting mechanism k 2 F(; K), the following statements are equivalent: (1) k is "-EPIC. (2) here exists a joint menu C 2 P f (K) such that " (; C) is well-de ned, and k is a Borel-measurable selection from " (; C), that is, k(t) 2 " (t; C) for all t 2. Proof. See Appendix. Furthermore, as long as " (; C) is well-de ned for some C 2 P f (K), the "-ex post menu design problem (P2) can be rewritten compactly: max max u( e k(t); t)(dt): C2P f (K) e k(t)2"(t;c) Now de ne the feasible individual-based "-EPIC mechanism set IC I " = k 2 F(; K) : k is "-EPIC. he individual-based "-EPIC mechanism design problem (P1 0 ) can also be stated compactly as max u(k(t); t)(dt): k2ic I "

15 Besides, de ne the equivalent mechanism set induced by a feasible joint menu C " (C) = k 2 F(; K) : k(t) 2 " (t; C) for all t 2 : It denotes the set of all measurable selections from " (; C) in F(; K) for a given menu C 2 P f (K). Next, de ne the overall equivalent mechanism set induced by the joint feasible menu set " = [ " (C). C2P f (K) herefore, the mutual characterization theorem for "-ex post implementation indeed suggests that IC I " = ". Now we are ready to see an important result on the strategic equivalence of individual-based "-EPIC mechanisms design and "-ex post menu design. heorem 1 (Delegation Principle for "-Ex Post Implementation). For every " 0, the following statements are true: (1) If k solves the contracting problem over individual-based "-EPIC mechanisms given by (P1 0 ), then C Q = n clf(k i (t i ) : t i 2 i g \ K solves the contracting problem over "-ex post menus given by (P2). (2) If C solves (P2), then k satisfying k (t) 2 argmax u( e k(t); t) e k(t)2"(t;c ) for all t 2 solves (P1 0 ). Moreover, the optimal objective values of the two problems are equal. Proof. See Appendix. his delegation principle indicates that "-ex post menu design is strategically equivalent to individual-based "-EPIC mechanism design rather than joint-based "-EPIC mechanism design. In addition, "-ex post menus only incorporate separate information as individual-based "-EPIC

16 mechanisms do rather than relative information as joint-based "-EPIC mechanisms do in specifying contract-pairs for each agent. 5 Comparative Analysis between Mechanism Design and Menu Design 5.1 Weak Dominance of Joint-based Mechanism Design An immediate concern after the delegation principle is whether joint-based mechanism can function better for PL than individual-based mechanism design and menu design. De nition 7 A joint-based "-ex post mechanism weakly (respectively, strictly) dominates an individual-based "-ex post mechanism or an "-ex post menu if the joint-based "-ex post mechanism makes PL (respectively, strictly) better o than the individual-based "-ex post mechanism or "-ex post menu, that is, the joint-based "-ex post mechanism yields at least as high an (respectively, a strictly) expected payo to PL as the individual-based "-ex post mechanism or. "-ex post menu. De nition 8 Joint-based "-ex post mechanism design weakly (respectively, strictly) dominates individual-based "-ex post mechanism design or "-ex post menu design if optimal individual-based "-ex post mechanism or optimal "-ex post menu is weakly (respectively, strictly) dominated by a joint-based "-ex post mechanism. By the delegation principle, individual-based "-EPIC mechanism serves as a bridge for us to connect all the procedures in contracting games and compare them. he core is to compare the joint-based "-EPIC mechanism design problem (P1) with the individual-based "-EPIC mechanism design problem (P1 0 ). For simplicity, rst de ne PL s expected payo by using direct mechanisms u(k) = u(k(t); t)(dt):

17 Also de ne IC J " = fk 2 F(; K) : k is "-EPIC.g IC I " = k 2 F(; K) : k is "-EPIC. hus, (P1) and (P1 0 ) can be rewritten compactly as below respectively: max u(k) k2ic J " and max u(k) k2ic I " We are also interested in when contracting games are decentralizable if the centralized mechanisms are allowed to be joint-based. Let i :! i be the projection function de ned by i (t 1 ; ; t n ) = t i. hen ((k i i ) i2n ) 2F (; K) is equivalent to k 2 F(; K). hen we de ne the projective joint-based "-EPIC mechanism set IC P " = (k i i ) i2n 2 F (; K) : k 2 IC I " : Observe that IC I " and IC P " are equivalent, i.e. they are in 1-1 correspondence with each other. Now consider a new mechanism design problem (P1 00 ): max U(k): k2ic P " one can view (P1 0 ) and (P1 00 ) as essentially equivalent problems. herefore, the collection of individual-based "-EPIC mechanisms is essentially equivalent to a subset of the collection of joint-based "-EPIC mechanisms. Optimal joint-based mechanism will make PL better o than the optimal individual-based mechanism and the optimal menu. Clearly, the weak dominance of joint-based "-EPIC mechanism design can be established thank

18 to the delegation principle. Proposition 2 For every " 0, joint-based "-EPIC mechanism design weakly dominates both individual-based "-EPIC mechanism design and "-ex post menu design. Compared with individual-based "-EPIC mechanisms, joint-based "-EPIC mechanisms suggest the relative information evaluation, that is, PL s contract-speci cation for each agent is on the basis of peer types (reports). She can refer to not only single agent s information communication but also all the other agents. With the relative information evaluation, PL may enhance her e ciency of decision making to deal with information asymmetry. Centralization of the right to specify contracts can take advantage of joint-based "-ex post mechanisms, in the sense of relative information evaluation, to make PL better o than decentralization in contracting. 5.2 Characterization of "-EPIC mechanisms We can also examine the di erence between joint-based and individual-based "-EPIC mechanisms in view of the concept "Weak Monotonicity," which is introduced by Bikhchandani et al (2006) to characterize deterministic dominant-strategy mechanisms. It is not technically demanding to extend this concept to ex post implementation. Weak monotonicity is a simple and intuitive condition on direct mechanisms. De nition 9 For every " 0, a joint-based mechanism k satis es "-Ex Post Weak Monotonicity ("-EPWM) if for each i 2 N ; t 2, v i (k(t 0 i; t i ); t 0 i; t i ) v i (k(t); t 0 i; t i ) v i (k(t 0 i; t i ); t) v i (k(t); t) "; for all t 0 i 2 i. An individual-based mechanism k satis es "-Ex Post Weak Monotonicity ("-EPWM) if for each i 2 N ; t 2 ; v i (k i (t 0 i); k i (t i ); t 0 i; t i ) v i (k i (t i ); k i (t i ); t 0 i; t i ) v i (k i (t 0 i); k i (t i ); t) v i (k i (t i ); k i (t i ); t) ";

19 for all t 0 i 2 i: Just as Bikhchandani et al (2006) suggest, the implication of weak monotonicity of a mechanism is as follows: if changing one agent s type (while keeping the types of other agents xed) changes the allocations and this agent s payo under the direct mechanism, then the resulting di erence in payo s of the new and original allocations evaluated at the new type of this agent must be no less than such di erence in payo s evaluated at his original type minus ". "-EPWM requires the mechanisms to be sensitive to changes in di erences in payo s. he "-EPWM condition is more intuitive and easier to check than "-EPIC. In fact, "-EPWM is the necessary condition for "-EPIC mechanisms 15. Proposition 3 For every " 0, if a joint-based or individual-based direct mechanism is "- EPIC, then it must satisfy "-EPWM. Proof. If a joint-based direct mechanism k is "-EPIC, then for each i 2 N ; t 2 ; v i (k(t); t) v i (k(t 0 i; t i ); t) "; (4) for all t 0 i 2 i. Also we have that for each i and t, v i (k(t 0 i; t i ); t 0 i; t i ) v i (k(t); t 0 i; t i ) "; (5) for all t 0 i 2 i. (5) and (6) yield that for each i and t, v i (k(t 0 i; t i ); t 0 i; t i ) v i (k(t); t 0 i; t i ) v i (k(t 0 i; t i ); t) v i (k(t); t) "; for all t 0 i 2 i. hus k satis es "-EPWM. he proof for individual-based mechanisms is similar. "-EPWM can also be thought of as a version of characterization of two types of "-EPIC mechanisms by this proposition. Let us recall the trick of introducing projection mapping in 15 As Bikhchandani et al (2006) show, it is also possible to make "-EPWM the su cient and necessary condition for "-EPIC in ordered domains of types.

20 previous analysis and compare (5) and (6). Intuitively, joint-based "-EPIC mechanisms turn out to be more sensitive to changes in di erences in payo s resulting from changing one agent s type than individual-based "-EPIC mechanisms, since the sensitivity in individual-based "-EPWM has been imposed the projection restriction on relative to that in joint-based "-EPWM. hus, more direct mechanisms may become "-EPIC and satisfy "-EPWM in the joint-based sense than in the individual-based sense. 5.3 Strict Dominance of Joint-based Mechanism Design Weak dominance cannot guarantee strict dominance. One may be interested in that jointbased "-EPIC mechanism design must make PL strictly better o than both individual-based "-EPIC mechanism design and "-ex post menu design. An immediate candidate for such a jointbased "-ex post mechanism could be some perturbation of the given individual-based "-EPIC mechanisms, which becomes a "pure" joint-based mechanism that preserves "-EPIC and brings higher expected payo to PL. 16 It is desirable to nd the relevant conditions on the preliminaries for strict dominance which are economically interesting. he following concepts capture some economically interesting intuition and are useful in provision of conditions for strict dominance of joint-based mechanism design. De nition 10 PL has strict con ict of interests with A i at type pro le t if u(k 0 ; t) > (resp: <) u(k; t) whenever v i (k; t) > (resp: <)v i (k 0 ; t): One important root cause of principal-agent problems is that two parties have con ict of interests. So it is reasonable to predict that PL may have strict con ict of interests with some agent at some type pro le. De nition 11 For each A i, v i is strictly ordered at type pro le t if v i (k; t) > v i (k 0 ; t) or v i (k; t) < v i (k 0 ; t) for each pair k 6= k 0 in K: Strictly-ordered preference of A i at type pro le t means that the A i cannot be indi erent between any di erent contract-pair given a type pro le t. he well-known single peaked 16 Moreover, if the optimal individual-based EPIC mechanism is k, and the candidate joint-based mechanism is k 0, the di erence between U(k 0 ) and U(k ) indicates at least how much the optimal joint-based mechanism will make the principal better o than the optimal individual-based mechanism and the optimal menu.

21 preference is a particular case of strictly-ordered preference at t. It is tractable to nd some desirable perturbation k 0 in nite contracting games, since there are only nitely many mechanisms. We now establish the rst set of su cient conditions for strict dominance of joint-based mechanism design in nite contracting games. De nition 12 A mechanism k i is nonconstant at type pro le t with respect to j if there exists at least one t 0 j such that (t0 j ; t j) 2 and k i (t 0 j ; t j) 6= k i (t j ; t j ). Proposition 4 For every " 0, suppose that the optimal solution to (P1 00 ) is k 2 IC P ". Let D; K, and be nite. For each t 2, (t) > 0. For some i and et = (et i ; et i ) 2, k i is nonconstant at et with respect to i. If (i) PL has strict con ict of interests with A i at type pro le et, and (ii) v i is strictly ordered at type pro le et, then joint-based "-EPIC mechanism design strictly dominates individual-based "-EPIC mechanism design or "-ex post menu design. Proof. First let bt i 2 farg max t 0 i v i (k i (t 0 i; et i ); k i(t 0 i; et i ); et)j(t 0 i; et i ) 2, k i (t 0 i; et i ) 6= k i (et i ; et i )g: Consider a mechanism k 0 in F(; K)nF (; K) de ned by k 0 (t) = f k (bt i ; et i ); k (t); t = et otherwise By this setting and hypothesis (ii), we know k 0 is "-EPIC and v i (k 0 (et); et) < v i (k (et); et): By hypothesis (i), u(k 0 (et); et) > u(k (et); et): Further, u(k 0 ) > u(k ).

22 hings could be more complicated in in nite contracting games. We now introduce a few more economically interesting concepts or hypotheses, so as to establish two sets of su cient conditions for strict dominance of joint-based mechanism design in in nite contracting games. De nition 13 A subset X of a vector space is star-shaped about zero if x 2 X for any x 2 X and any 2 [0; 1]. Star shape about zero is more general than convexity containing 0 (identity element of addition). It implies that any vector in X after scale shrinkage still lies in X. It is worth noting that it is still independent of order structures. Example 3 In example 1 case 3), the feasible joint contract set K for a single object is starshaped about zero. De nition 14 Given D; K, and are all subsets of vector spaces, v i is homogeneous in k of degree z i if for some integer z i, v i (k; t) = z i v i (k; t) for each (k; t) 2 K and > 0. he homogeneity of A i s payo with respect to contract-pairs re ects that there is some scale e ect (or scale invariance) on A i s payo with respect to contract-selection. An immediate example is linearity of v i in k. Example 4 In a canonical quasilinear environment with pure adverse selection and voluntary participation, k i = (k i1 ; k i2 ); where k i1 2 R denotes the alternative for i, and k i2 2 R denotes the transfer to i. If v i (k i ; t) = g(t)k i1 + k i2 ; where there exists some continuous function g :! R; v i is homogeneous in k i of degree 1. De nition 15 u has return to scale shrinkage in k if u(k; t) > u(k; t) for each (k; t) 2 K and each 2 (0; 1). Return to scale shrinkage of u in k means that the proportional shrink of contracts leads to increase in PL s payo. Example 5 Suppose k = (x; s) 2 [0; n] [0; 1] n satisfying x P s i is the pro le of public good provision and transfer from the agents to PL. Let u(k; t) = 2x + P s i. he marginal cost

23 of providing public good is greater than marginal bene t of collecting money from the agents in absolute value. hus u has return to scale shrinkage in k. Proposition 5 For every " 0, suppose that the optimal solution to (P1 00 ) is k 2 IC P ". Let D; K, and be subsets of vector spaces. is also normed. K is star-shaped about zero. If (i) for each i; k; t, v i (k; t) v i (k i ; t) and v i is homogeneous in k of degree z i 2 (when " > 0, z i needs to be positive), and (ii) u has return to scale shrinkage in k, then joint-based "-EPIC mechanism design strictly dominates both individual-based "-EPIC mechanism design and also "-ex post menu design. Proof. First consider a mechanism k 0 de ned by k 0 (t) = t ktk k (t) for each t 2 : where kk denotes the norm on, and t is some real constant in (0; 1) such that k 0 (t) 2 K, for each t 2. Easy to see that k 0 is also "-EPIC by hypothesis (i). By hypothesis (ii), u(k 0 (t); t) = u( t ktk k (t); t) u(k (t); t); where the strict equality holds only when t = 0. hus, u(k 0 ) > u(k ). Proposition 4 requires that there is no allocation externality. Proposition 5 below can evade this requirement when the solution to individual-based mechanism design problems is semi-strict "-EPIC. 17 De nition 16 A direct mechanism k is semi-strict "-ex post incentive compatible (Semi- 17 We borrow this idea from Bergemann and Morris (2011).

24 Strict "-EPIC) if for each i; t; v i (k(t); t) > v i (k(t 0 i; t i ); t) "; whenever t 0 i is such that (t0 i ; et i ) 2 and k(t 0 i ; t i) 6= k(t i ; t i ). Semi-strict "-EPIC mechanisms can be ensured in some economically natural applications. For instance, if for each i and t, v i is strictly ordered at type pro le t, then any "-EPIC mechanisms must be semi-strict "-EPIC. Proposition 6 For every " 0, suppose that the optimal solution to (P1 00 ) is k 2 IC P ". If (i) k is semi-strict "-EPIC, and (ii) there exists a sequence fk l g l in F(; K)nF (; K) such that as l! 1; k l (t)! k (t) for each t, and for some N; u(k l ) > u(k ) whenever l N, then joint-based "-EPIC mechanism design strictly dominates both individual-based "-EPIC mechanism design and also "-ex post menu design. Proof. Pick the sequence fk l g l2n in F(; K)nF (; K) as described in hypothesis (ii). Recall that v i (; t) is continuous for all i and t. hus for some N c, when l N c, the semi-strict "-EPIC optimal solution to (P1 00 ) implies that v i (k l (t); t) > v i (k l (t 0 i ; t i); t), whenever t 0 i is such that (t 0 i ; et i ) 2 and k(t 0 i ; t i) 6= k(t i ; t i ). Clearly, k l 2 IC J " nic P ". Now pick m = maxfn; N c g. u(k m ) > u(k ) by hypothesis (ii). If the optimal solution to (P1 00 ) k can be tended by a sequence of "pure" joint-based direct mechanisms that are not equivalent to individual-based direct mechanisms, there exists some element in the sequence that can inherit the "-EPIC property from the semi-strict "-EPIC k. herefore, k must be dominated at least by that element from the viewpoint of PL. Moreover, if the elements of the sequence will eventually bring higher expected payo to PL, (P1) strictly dominates (P1 0 ) and (P2). We now establish an corollary to Proposition 5 in vector space structures. Corollary 1 For every " 0, suppose that the optimal solution to (P1 00 ) is k 2 IC P ". Let D; K, and be subsets of vector spaces. is also normed. K is star-shaped about zero. If

25 (i) k is semi-strict "-EPIC, and (ii) u has return to scale shrinkage in k, then joint-based "-EPIC mechanism design strictly dominates both individual-based "-EPIC mechanism design and also "-ex post menu design. Proof. First consider a sequence of mechanisms (k l ) l2n de ned by k l (t) = ( t;l ktk) 1 l k (t); for each t 2 ; where kk denotes the norm on, and for each t and l, t;l is some real constant in such that k l (t) 2 K. Note that ( t;l ktk) 1 l 2 (0; 1). Clearly, fk l g l F(; K)nF (; K), because K is star-shaped about zero. Moreover, for each t, as l! 1, k l (t)! k (t). By hypothesis (ii), u(k l (t); t) = u( t;l ktk) 1 l k (t); t) u(k (t); t); where the strict equality holds only when t = 0. Hence, u(k l ) > u(k ). herefore, Proposition 5 applies here. 5.4 Quasi-linear Environments and Strategic Dominance he quasilinear environment is a special and much studied class of environments in mechanism design theory. It is quite reasonable and tractable in most cases. We can characterize or derive both optimal joint-based and individual-based EPIC mechanisms and closely examine the strategic dominance in quasilinear environments. In a classic quasilinear environment, t i 2 i = [t i ; t i ]. t has a joint distribution function F :! [0; 1] and an associated joint density function f :! R + with respect to Lebesgue measure. For each i 2 N, there are a marginal distribution function F i : i! [0; 1]. Contracts consist of assignments and transfers. he assignment is denoted by x 2 X, where X is a partialordered set of a metric space. he transfer from PL to A i is denoted by s i 2 R. Let Borel measurable function x i :! X (resp. x i : i! X ) specify to A i the assignments for each type report pro le of the agents (resp. for each type report of A i ). Let Borel measurable function