Truthful Equilibria in Generalized Common Agency Models
TTruthful Equilibria in Generalized CommonAgency Models
Ilias Boultzis ∗ Keywords: common agency, truthful equilibria, externalities, efficiency JELclassification: C72, D44, D43, D72
Abstract
In this paper I discuss truthful equilibria in common agency models.Specifically, I provide general conditions under which truthful equilibria areplausible, easy to calculate and efficient. These conditions generalize similarresults in the literature and allow the use of truthful equilibria in noveleconomic applications. Moreover, I provide two such applications. The firstapplication is a market game in which multiple sellers sell a uniform goodto a single buyer. The second application is a lobbying model in whichthere are externalities in contributions between lobbies. This last exampleindicates that externalities between principals do not necessarily preventefficient equilibria. In this regard, this paper provides a set of conditions,under which, truthful equilibria in common agency models with externalitiesare efficient.
The common agency model is a game in which many principals share a commonagent. Economists apply this model in many areas of economic research likelobbying, industrial organization, public economics e.t.c.Common agency models often have many equilibria. However, truthful equi-libria are probably the most popular among them. This type of equilibria was ∗ Department of Economics, Athens University of Economics and Business, e-mail: [email protected]. I would like to thank the participants of the 2019CRETE conference for their valuable comments. I would also like to thank Apos-tolis Philippopoulos for our discussions, from an early stage of this paper andKaterina Katsandredaki for her assistance in the editing of this paper. The usualdisclaimer applies. 1ntroduced by Bernheim and Whinston (1986) for quasi linear utility functionsand was generalized to all utility functions by Dixit et al. (1997). Specifically,Dixit et al. (1997) argued that truthful equilibria are focal because they havethree important properties: they are easy to calculate, Pareto efficient and plau-sible. The last property means that the best response set of the principals alwayscontains a strategy which is consistent with truthful equilibria (truthful strategy).However, the analysis by Dixit et al. (1997) was motivated by lobbying games.Following this motivation, they based their results on a general setting associatedwith lobbying models. This setting consists of three key assumptions. First, theutility of the principals (lobbyists) decreases with the bids (contributions) theyoffer to the agent (politician). Second, the utility of the principals depends onlyon their own bids. Third, the utility of the agent is increasing in all bids offeredby the principals.Nevertheless, many applications of common agency do not fit this restrictedframework. Thus, a question arises. Is it possible to find a broader set of conditionsunder which truthful equilibria can be used? In this paper, I attempt to answerthis question. Specifically, I provide general conditions under which the results ofDixit et al. (1997) survive. These conditions allow the use of truthful equilibria innovel economic applications.In this paper I consider two such applications. The first application is a marketgame in which a group of principals (sellers), sells a uniform good to a single agent(buyer). The sellers move first and present the buyer with a bid (price) whichis conditioned on the amount that the buyer wishes to buy. Then, the buyerdecides on the quantity he buys from each seller. This situation is the reverseof the lobbying model by Dixit et al. (1997), since typically, the utility of thesellers increases, while the utility of the buyers decreases following a rise in prices.The properties of truthful equilibria in such market games were established byBernheim and Whinston (1986) for quasi linear utility functions. Here I followDixit et al. (1997) and generalize them to all utility functions.The second application considers externalities in bids, among principals. Theseexternalities occur naturally when principals are interrelated in other ways besidessharing a common agent. For example, think of a federal country in which the2tates (principals) lobby the central government (agent) for a transfer. Moreover,assume that the states finance voluntarily a public good like public safety. In thiscase, all states benefit from public safety spending in the other states. However,this spending depends on the available resources of each state. In turn, theseavailable resources decrease with lobbying. Thus, the welfare in each state dependson lobbying expenditures in all states. Therefore, externalities in bids, in this casespending for lobbying, emerge.The use of truthful equilibria in the two applications above is not equallyintuitive. Specifically, in market games the setting resembles the model of Dixitet al. (1997). Thus, the fact that truthful equilibria retain their properties doesnot come as a surprise. However, externalities in bids is a different story. In suchmodels the literature has identified efficiency failures and thus the efficiency oftruthful equilibria is less expected. In particular, externalities in bids can leadto inefficiency because of a possible prisoners’ dilemma. The set of conditions Iprovide in this paper imply that this prisoners’ dilemma need not always appear.Thus, there exist applications with externalities in bids, in which truthful equilibriahave all three properties identified by Dixit et al. (1997), including efficiency.In this regard, my model belongs in the strand of literature that exploresthe robustness of truthful equilibria to various extensions of the original com-mon agency model. Dixit et al. (1997) pioneered this literature by extending themodel of Bernheim and Whinston (1986) to general utility functions. Other im-portant examples of this literature include Bergemann and V¨alim¨aki (2003) whoconsider dynamic common agency, Prat and Rustichini (2003) who discuss multi-ple agents and Martimort and Stole (2009b) who consider asymmetric information.Furthermore, my paper relates to the literature discussing common agency withexternalities in bids. Examples of this literature are Peters (2001), Martimort andStole (2002), Peters and Szentes (2012), Szentes (2014) and Galperti (2015). Thepapers in this strand of literature discuss models with asymmetric informationand observe that externalities in bids can lead to inefficient equilibria. Followingthis observation, they allow bids to depend on the bidding strategies of the other See Peters (2001) and Martimort and Stole (2002).3rincipals in order to restore efficiency. These papers relate to my work, to theextent that their results also apply to symmetric information models. In this re-spect, my contribution is that truthful equilibria can be efficient, even if bids donot depend on other bidding strategies.The rest of the paper is organised as follows. Section 2 describes the mainmodel, section 3 presents the key results and section 4 investigates a variation ofthe main model. Section 5 discusses the general issue of efficiency of equilibriaand the relationship between my paper and the existing literature on externali-ties. Section 6 considers economic applications and section 7 concludes. Finally,appendix A contains the main proof of the paper, while appendix B, which is notintended for publication, contains the rest of the proofs along with some examplesand calculations.
Before I continue with the details of the model, let me introduce the notation thatI use in the rest of the paper. The index i runs from 1 to n. Furthermore, exceptwhen otherwise stated, I use Latin and Greek letters in the following manner.Consider for example the lower-case letter “ x ” . Then, x i is a real number, x is the vector of all x i , x − i is the vector containing all members of x except x i , ˜ x = (cid:80) i x i and ˜ x − j = (cid:80) i (cid:54) = j x i . Moreover, I use the symbol x i ( · ) to describe afunction x i : Z → R, such that x i = x i ( z ) , for a given set Z . The symbols x ( · )and x − i ( · ) describe the respective vectors of functions. Finally, if x , y are twovectors, then x ≥ y means that x i ≥ y i for all i, while x > y means that x i ≥ y i for all i and there exists at least one i, such that x i > y i . I turn now to the model. Consider a common agency model with one agentand n principals. Following Dixit et al. (1997) I discuss here what is known as apublic common agency model . I depart from this assumption in section 4. In public common agency models the principals condition their bids on theentire action of the agent, while in private common agency, the principals4 gent.
The agent chooses an element a from the set A . The set A reflectsbudget, institutional or other constraints that depend on specific applications. Forexample, if the agent is a government choosing a tax rate, the set A is the interval[0 , A is a subset of R n + . Henceforth, I refer to a as theagent’s action.The utility function of the agent is a function: u : A × R n → R such that u o = u ( a, b ) . The vector b ∈ R n , is the vector of bids that the principals submit to the agentin order to influence the choice of a. This utility function is strictly monotonouswith respect to all bids and continuous with respect to all its elements.
Principals.
On the other hand, each principal chooses a bidding function: b i : A → R such that b i = b i ( a ) , in order to influence the agent. These bidding functions meet appropriate restric-tions. Specifically, there exist two functions b i : A → R and b i : A → R, which areuniformly bounded above and below by b max , b min ∈ R respectively and satisfythe inequality b i ( a ) ≥ b i ( a ) for all a ∈ A. These functions define feasible bids:
Definition 1.
Feasibility : A bid b i ∈ R is feasible relative to a ∈ A , if b i ∈ [ b i ( a ) , b i ( a )]. Moreover, a bidding function b i ( · ) is feasible, if b i ( a ) is fea-sible relative to a, for all a ∈ A .Additionally, the vector b ∈ R n is feasible relative to a ∈ A if all b i are feasiblerelative to a . In this case I say that the pair ( a, b ) is feasible. A feasible pair ( a, b )is symmetric if b i = b j for all i, j. Similarly, the vector of bidding functions b ( · )is feasible, if all its elements are feasible. Moreover, if b ( · ) is feasible and a ∈ A, Isay that ( a, b ( · )) and ( a, b i ( · )) are feasible. A feasible pair ( a, b ( · )) is symmetric condition their bids only on a part of the agent’s action (i.e. the part theyobserve). For more on the meaning of these terms see Martimort and Stole(2009a). 5f b i ( · ) = b j ( · ) for all i, j. Feasibility restrictions in bids reflect application specific constraints. For ex-ample, if the principals are lobbies offering campaign contributions, the bids mustbe positive and not exceed the budget constraint of the lobby. Likewise, if theprincipals are sellers and the bids are selling prices, the bids should be greaterthan the cost of acquisition and smaller than the buyer’s reservation price.Now I turn to the utility functions of the principals. Specifically, the utilityfunction of principal i is a function: u i : A × R n → R such that u i = u i ( a, b ) . This utility function is strictly monotonous with respect to own bids b i and con-tinuous with respect to both a and b . Timing.
Finally, the timing of the model is standard. There are two stages.In stage one, the principals submit simultaneously their bidding functions. Instage two, the agent chooses his action and the bids are realised.Following the analysis above, my model is fully described by the number ofprincipals n, the n + 1 utility functions, the set A and the restrictions in bids( b i ( · ) , b i ( · ) , b max , b min ) . Henceforth, I use the term game, whenever I refer to acommon agency model defined in such a way.This game extends the model by Dixit et al. (1997) in two ways. First, theutility of each principal also depends on the bids of all other principals. Thus,my model allows for externalities among principals. Second, I make no priorassumptions, regarding the effect of bids on utility, other than monotonicity. Theseextensions broaden the range of applications to which truthful equilibria apply. Idiscuss these issues again in 2.4 which provides further specification of the modeland in section 6 which discusses economic applications.Let me now turn to the equilibrium of the game.6 .2 Equilibrium
The definitions of best response and equilibrium below extend the respective def-initions by Dixit et al. (1997) . Definition 2.
Best response : A feasible bidding function b i ( · ), belongs inthe best response set of principal i , to the feasible bidding functions b − i ( · ) of theother principals, if:There exists an a ´ ∈ arg max a ∈ A u ( a, b ( a )), such that there does not exist a feasiblepair ( a ∗ , b ∗ i ( · )), such that u i ( a ∗ , b ∗ i ( a ∗ ) , b − i ( a ∗ )) > u i ( a ´ , b ( a ´)) and a ∗ ∈ arg max a ∈ A u ( a, b ∗ i ( a ) , b − i ( a )). Definition 3.
Equilibrium : A feasible pair ( a o , b o ( · )) is an equilibrium if: a) a o ∈ arg max a ∈ A u ( a, b o ( a )) and b) for all i , there does not exist a feasible pair ( a ´ , b i ( · )), such that a ´ ∈ arg max a ∈ A u ( a, b i ( a ) , b o − i ( a ))and u i ( a ´ , b i ( a ´) , b o − i ( a ´)) > u i ( a o , b ( a o )).Let me now turn to the notion of truthful equilibrium. Truthful equilibria refine the equilibria described in definition 3. They were in-troduced by Bernheim and Whinston (1986) and were generalized by Dixit etal. (1997). In truthful equilibria the principals submit truthful bidding functions.These functions exactly reflect changes in the utility of the principals that followfrom changes in the actions of the agent. Thus, truthful bidding functions revealthe true preferences of the principals. Definitions 4 and 5 below adapt these ideasto my setting.Let ( a, b ( · )) be a feasible pair and u ∗ i ∈ R . Consider the equation u ∗ i = u i ( a, φ i , b − i ( a )) , with respect to φ i . Since b ( · ) is a vector of feasible bidding func-tions, b − i ( a ) exists for all a ∈ A . Furthermore, because the utility of the principals On the definitions 2 and 3 see also Ko (2011) and Ko (2017).7s monotonous in own bids, this equation always has a unique solution. This solu-tion defines a function φ i : A → R such that φ i = φ i ( a ; u ∗ i , b − i ( · )). Then, I definetruthful responses as follows: Definition 4.
Truthful response : A bidding function b Ti : A → R , is atruthful response of principal i to the feasible bidding functions b − i ( · ) of the otherprincipals, relative to the constant u ∗ i , if : a) b Ti = b i ( a ) if u i ( a, b i ( a ) , b − i ( a )) < u ∗ i φ i ( a ; u ∗ i , b − i ( · )) if u i ( a, b i ( a ) , b − i ( a )) ≤ u ∗ i ≤ u i ( a, b i ( a ) , b − i ( a )) b i ( a ) if u ∗ i < u i ( a, b i ( a ) , b − i ( a ))and u i ( · ) is strictly decreasing in own bids, or b) b Ti = b i ( a ) if u i ( a, b i ( a ) , b − i ( a )) > u ∗ i φ i ( a ; u ∗ i , b − i ( · )) if u i ( a, b i ( a ) , b − i ( a )) ≥ u ∗ i ≥ u i ( a, b i ( a ) , b − i ( a )) b i ( a ) if u ∗ i > u i ( a, b i ( a ) , b − i ( a ))and u i ( · ) is strictly increasing in own bids.Definition 4 states that truthful responses are equal to the expression φ i ( a ; u ∗ i , b − i ( · ))except when this expression violates lower or upper feasibility bounds. In suchcases the truthful responses are equal to these bounds. Therefore, truthful re-sponses are by construction feasible bidding functions .Now I can turn to the definition of truthful equilibrium. Definition 5.
Truthful equilibrium : Let ( a o , b o ( · )) be an equilibrium ofthe game and u o = u ( a o , b o ( a )) be the vector of equilibrium utility levels of the Ko (2011) explains the advantages of definition 4 when compared to b Ti = min { b i ( a ) , max { b i ( a ) , φ i ( a ; u i , b − i ( · )) }} which corresponds to the definitionprovided by Dixit et al. (1997). 8rincipals. This equilibrium is truthful, if for all i , the equilibrium bidding func-tion b oi ( · ) is a truthful response of principal i to the equilibrium bidding functions b o − i ( · ) of the other principals, relative to his equilibrium utility level u oi .Let me clarify the notion of truthful equilibrium. Consider any feasible pair( a, b ( · )) and let u = u ( a, b ( a )) , be the vector of corresponding utility levels. Then,definition 4 determines the truthful response, for each principal, to the biddingfunctions b − i ( · ) of the other principals, relative to u i . For a given a, this operationdefines a mapping from the n-dimensional space of feasible bidding functions toitself. Now consider an equilibrium pair ( a o , b o ( · )) and let u o = u ( a o , b o ( a o )) . If b o ( · ) is a fixed point in the mapping above, then it is a truthful equilibrium .Next, I discuss the structure of the game. Here, I provide some additional assumptions that often characterize applications.
Assumption A.
Opposing monotonicity
A1. Lobbying.
The utility of all principals is strictly decreasing in own bids andthe utility of the agent is strictly increasing in all bids. Moreover, b i ( a ) = b min , for all i and a ∈ A. The existence of such a fixed point deals with the issue of infinite regress thatcan appear in common agency with externalities. For the problem of infiniteregress in common agency see Peters (2001), Martimort and Stole (2002) andmore recently Szentes (2014) and Galperti (2015). In these papers the biddingfunctions explicitly depend on other bidding functions. Therefore, infiniteregress can appear because the bidding function of principal i depends on thebidding function of principal j, which in turn depends on the bidding functionof principal i and so on. My setting does not allow for explicit dependence onother bidding functions. However, the principals form guesses about the otherbidding functions which also depend on the guesses that the other principalsform and so on. 9
2. Market.
The utility of all principals is strictly increasing in own bids andthe utility of the agent is strictly decreasing in all bids. Moreover, b i ( a ) = b max , for all i and a ∈ A. Assumption A1 is often satisfied in lobbying models. In these models the prin-cipals are lobbies and their bids are usually campaign contributions to politicians.In such a case, the politicians like receiving contributions while the lobbies dislikepaying them. Moreover, contributions must be non-negative and therefore, thecommon lower bound b min is zero. Assumption A2 is more relevant in marketgames, in which the principals are sellers of a homogeneous good and their bidsare selling prices. In this case, the sellers like high prices while the buyer dislikesthem. Furthermore, the upper bound of the bids is the buyer’s reservation price.Henceforth, when I refer to assumptions A1 and A2, I use the terms lobbyingand market monotonicity respectively. Assumption B.
No externalities.
The utility of the principals takes the form u i : A × R → R such that u i = u i ( a, b i ).Assumption B describes a special case without externalities in bids. The com-bination of lobbying monotonicity and no externalities defines the lobbying modeldiscussed by Dixit et al. (1997). Assumption C.
Conflict of interests at ( a, b ) . Let ( a, b ) be a feasible pair. If there exists a feasible pair ( a, b ´) such that u ( a, b ´) >u ( a, b ) then there exists an i such that u i ( a, b ´) < u i ( a, b ) and if there exists afeasible pair ( a, b ´) such that u ( a, b ´) > u ( a, b ) then u ( a, b ´) < u ( a, b ) . Assumption C states that for a given agent’s action, it is impossible to changebids, in a way that makes both the principals and the agent better off. There-fore, this assumption introduces conflict of interests between principals and agent,over bids. Assumption C is implied by opposing monotonicity in games withoutexternalities. However, in games with externalities this is not always the case.10n order to explain the role of externalities in this issue, let me consider agame with lobbying monotonicity. First, assume that there are no externalities.Also, fix the agent’s action at a certain level and consider a change in bids whichmakes the agent better off. Such a change requires that at least one bid ( b i ) in-creases. Then, since u i = u i ( a, b i ) , the utility of at least one principal decreases.Thus, these changes are consistent with conflict of interests. Alternatively, con-sider a similar game with externalities. In this case, the utility of principal i is u i = u i ( a, b i , b − i ) . Furthermore, assume that the utility of all principals is increas-ing in all the elements of b − i . Also, assume that all contributions increase. Then,two conflicting effects emerge. On the one hand, the increase in own bids has anegative effect on the utility of the principals. On the other hand, the increase inthe other bids has a positive effect on the principals. If the positive cross effectdominates the negative own effect, for all principals, then conflict of interests isviolated. In such a case, an increase in all bids increases the utility of all principalsand the agent. Assumption C appropriately restricts externalities to disallow suchsituations. This assumption along with assumption D that follows guarantee theefficiency of truthful equilibria in games with externalities. Assumption D.
Deep pockets
D 1.
Weak deep pockets at ( a o , b o ( · )) . Let ( a o , b o ( · )) be a truthful equilibrium and ( a ∗ , b ∗ ) be a feasible pair such that u ( a ∗ , b ∗ ) ≥ u ( a o , b o ( a o )) and u ( a ∗ , b ∗ ) ≥ u ( a o , b o ( a o )) with at least one strict in-equality, then u ( a o , b o ( a o )) ≥ u ( a ∗ , b o ( a ∗ )). D 2.
Strong deep pockets
There exists u i ∈ R such that: D 2.1.
If the utility of all principals is strictly decreasing in own bids then u i ( a, b i ( a ) , b − i ( a )) = u i ≤ u i ( a, b ( a )) for all i , a ∈ A and feasible b − i ( · ). D 2.2.
If the utility of all principals is strictly increasing in own bids then u i ( a, b i ( a ) , b − i ( a )) = u i ≤ u i ( a, b ( a )) for all i , a ∈ A and feasible b − i ( · ).The term “deep pockets” is due to Ko (2011) who uses a similar assumption.Furthermore, Dixit et al. (1997) also employ a version of strong deep pockets.11pecifically, Dixit et al. (1997) assume that there is a subsistence utility level u i and define b i ( · ) implicitly through u i ( a, b i ( a )) = u i . If a game satisfies strong deeppockets then it also satisfies weak deep pockets. However, as I show later on,many applications satisfy weak deep pockets directly. These applications includeall games without externalities but also a number of games with externalities.Definition 6 introduces some terms that I use in assumptions E and F below. Definition 6.
Game structure: a) A game is differentiable if all utility functions are differentiable with respectto all bids. b) A game is cumulative if the utility functions of the agent and the principalsare as follows: u : A × R → R such that u = u ( a, ˜ b ) and u i : A × R → R such that u i = u i ( a, b i , ˜ b − i ) for all i. c) A game exhibits negative externalities if it is differentiable, cumulative andthe utility of all principals is strictly decreasing in the total bids of the other prin-cipals (˜ b − i ). d) A game exhibits positive externalities if it is differentiable, cumulative andand the utility of all principals is strictly increasing in the total bids of the otherprincipals (˜ b − i ) . e) A game is symmetric if u i ( · ) = u j ( · ) , b i ( · ) = b j ( · ) and b i ( · ) = b j ( · ) for all i, j. f ) A game is quasi-concave if it is cumulative and the utility functions of allprincipals are quasi-concave with respect to own and other bids.
Assumption E.
Small externalities
Either the game exhibits negative externalities and (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂u i ∂b i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) > (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂u i ∂ ˜ b − i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) for all i and allfeasible ( a, b ) or the game exhibits positive externalities and (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂u i ∂b i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) > ( n − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂u i ∂ ˜ b − i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) for all i and all feasible ( a, b ). 12ssumption E is a special case of conflict of interests. Specifically, small exter-nalities describe a situation in which the effect of own bids appropriately dominatesthe respective cross effects. In this respect, think of ( n − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂u i ∂ ˜ b − i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) as the totalcross effect and of (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂u i ∂ ˜ b − i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) as the average cross effect. These two effects coincidewhen n = 2. As it turns out, this restriction in externalities achieves the conflictbetween agent and principals which is necessary to satisfy assumption C. Assumption F.
Symmetric negative externalities
The game is symmetric, quasi concave and exhibits negative externalities.Proposition 1 that follows explains the relationship between assumptions A-F.
Proposition 1.(i)
Strong deep pockets imply weak deep pockets at all truthful equilibria. (ii)
The combination of opposing monotonicity and no externalities, implies con-flict of interests at all feasible pairs ( a, b ) and weak deep pockets at all truthfulequilibria. (iii)
The combination of small externalities and lobbying monotonicity implies con-flict of interests at all feasible pairs ( a, b ) and weak deep pockets at all truthfulequilibria. (iv)
The combination of symmetric negative externalities and lobbying monotonic-ity implies conflict of interests at all symmetric pairs ( a, b ) and weak deep pocketsat all symmetric truthful equilibria.Proof: See appendix B.1.1. Also, Dixit et al. (1997) prove a part of (ii) by show-ing that lobbying monotonicity and no extermalities imply conflict of intersts andweak deep pockets during their proof of the efficiency of truthful equilibria .I turn now to the main results. For their proof see Dixit et al. (1996)13
Results
Dixit et al. (1997), like Bernheim and Whinston (1986) before them, argue thattruthful equilibria are focal, because they share three key properties. Namely,truthful equilibria are plausible, easy to calculate and efficient .Dixit et al. (1997) arrive at this result under the assumptions of lobbyingmonotonicity and no externalities. Here I generalize their argument by providinga broader set of conditions under which it is valid. The four propositions thatfollow achieve this task. Proposition 2.
Plausibility
Consider a game that exhibits opposing monotonicity. Then, the best response setof principal i to the bidding functions of the other principals b − i ( · ) always containsa truthful response.Proof: see appendix B.1.2. Proposition 3.
Calculation
Consider a game that exhibits opposing monotonicity. A. If the feasible pair ( a o , b o ( · )) is an equilibrium then (Ai) and (Aii) below aretrue: (Ai) a o ∈ arg max a ∈ A u ( a, b o ( a )) (Aii) For all i, lobbying monotonicity implies that u ( a o , b o ( a o )) = max a ∈ A u ( a, b min , b − i ( a ))and market monotonicity implies that u ( a o , b o ( a o )) = max a ∈ A u ( a, b max , b − i ( a )) B. Let b o ( · ) be a vector of feasible bidding functions, such that for all i, b oi ( · ) is atruthful response to b o − i ( · ) , relative to u oi ∈ R. Also let a o ∈ A be an agent’s actionsuch that the pair ( a o , b o ( · )) is feasible and satisfies (Ai) and (Aii) above. Then if(Bi) and (Bii) below are true ( a o , b o ( · )) is a truthful equilibrium of the game. A part of the literature expresses doubts regarding the relevance of truthfulequilibria. In this respect Kirchsteiger and Prat (2001) provide an experimen-tal argument and Martimort and Stole (2009b) provide a theoretical argumentagainst truthful equilibria. 14
Bi) u o = u ( a o , b o ( a o )) (Bii) There does not exist an a ∈ A such that the pair ( a, b o ( · )) is feasible, satisfiesconditions (Ai) and (Aii) and yields u ( a, b o ( a )) > u o . Proof: See appendix B.1.3.
Lemma to proposition 3B
If the game satisfies strong seep pockets then condition (Bii) is always satisfied.Proof: See the proof of part (i) of proposition 1 in appendix B.1.1.These results are true regardless of other characteristics of the game i.e. theexistence of externalities. In more detail, proposition 2 states that the principalsstand to loose nothing from responding truthfully to any bidding function chosenby the other principals. Also, the intuition behind proposition 3A is standard.In particular, principal i submits a bid that matches the agent’s outside option.Principal i has no motive to improve his bid any further.Proposition 3A holds for all equilibria. Proposition 3B states that the converseof 3A holds only for truthful equilibria and only under certain conditions. In orderto explain how proposition 3B works let me give two examples: First, considera game with two principals with utility functions u i = a − b i and an agent withutility function u = b + b . Moreover, assume that a ∈ [0 ,
1] and b i ∈ [0 , a ] . This game satisfies strong deep pockets since u i ( a, b i ) ≥ a, b i ) ,b i ( a ) = a and u i ( a, b i ( a )) = 0 for all a ∈ A. Moreover, in this game the biddingfunctions b i = 0 for all a ∈ [0 ,
1] are truthful responses to each other relative to u oi = 1 and satisfy conditions (Ai) and (Aii) for all a ∈ [0 ,
1] . Then, proposition3B implies that only a = 1 for which u i = 1 = u oi is part of a truthful equilibriumwith equilibrium utility u oi = 1 . Second, consider an example in which condition (Bii) fails. In particular con-sider a game in which the utility of the principals and the range of a are as inthe previous example, while b i ∈ [0 , ] and u = (cid:26) b + b − a if 0 ≤ a ≤ b + b − < a ≤ b oi ( a ) = (cid:26) a if 0 ≤ a ≤ if < a ≤ u oi = 0 . Additionally, the pair ( a, b o ( · ))15atisfies conditions (Ai) and (Aii) for all a ∈ [0 ,
1] and condition (Bi) for all a ∈ [0 , ] . However, condition (Bii) fails because for example, for a = 1 , u i (1 , b oi (1)) =0 . > u oi = 0 . Following this failure, the bidding functions b o ( · ) do not yield atruthful equilibrium. In order to verify this fact, consider a deviation by one ofthe principals to the truthful biding function with respect to u i = 0 . . Proposition 3A and especially proposition 3B are very helpful in the calculationof truthful equilibria. In this respect, I provide examples in section 6 and appendixB.3.Now I turn to efficiency.
Proposition 4.
Efficiency
Consider a game which exhibits conflict of interests and weak deep pockets at( a o , b o ( · )) , which is a truthful equilibrium of this game. Then, there does notexist a feasible pair ( a ∗ , b ∗ ), such that u ( a ∗ , b ∗ ) ≥ u ( a o , b o ( a o )) and u ( a ∗ , b ∗ ) ≥ u ( a o , b o ( a o )), with at least one strict inequality.Proof: See appendix AProposition 4 states that under certain conditions, truthful equilibria imple-ment an allocation which is Pareto efficient for all participants of the game (princi-pals and agent). Definition 7 and proposition 5 below summarize the results so far. Definition 7.
Validity
A truthful equilibrium of a game is valid, if it satisfies propositions 3 and 4 andthe game satisfies proposition 2.
Proposition 5.
Results a) All truthful equilibria are valid in all games that satisfy one of the following: (i) lobbying monotonicity and no externalities (ii) market monotonicity and no externalities For similar results in models with quasi linear utility functions see also Bern-heim and Whinston (1986), Laussel and Le Breton (2001), Martimort and16 iii) lobbying monotonicity and small externalities b) Symmetric truthful equilibria are valid in games that satisfy symmetric nega-tive externalities and lobbying monotonicity.Proof: Follows directly from propositions 1-4.Part (a-i) of proposition 5 restates the argument by Dixit et al. (1997) infavour of truthful equilibria in the standard model, while parts (a-ii), (a-iii) and(b) generalize this argument in different settings. The intuition behind proposition5 is straightforward for the case of market monotonicity and no externalities. Thiscase is the reverse of the model by Dixit et al. (1997) and thus its motivation issimilar. In the case of small externalities the intuition is also simple. Specifically, ifthe externalities are small they have no effect. I discuss this issue further in section5. For the case of symmetric externalities I provide examples that highlight therole of both symmetry and quasi-concavity in achieving conflict of interests, inappendix B.2.2.Proposition 5 describes 4 general settings in which truthful equilibria are rel-evant. However, other such settings might also exist. In this respect, interestedresearchers can check whether a specific application satisfies any or all of proposi-tions 2-4 directly.In section 6, I discuss economic economic applications of proposition 5. Now,I turn to a variation of the main model.
The analysis so far implies that the principals condition their bids on the entireagent’s action. For example, in a market game, the principals condition their sell-ing prices, both on the quantity they sell themselves to the agent, but also on thequantity all other principals sell to the agent. In section 6, I consider an exampleStole (2003), Segal and Whinston (2003) and Chiesa and Denicol`o (2009). Dasgupta and Maskin (2000) also find that small externalities lead to effi-cient equilibria in their study of efficient equilibria in Vickrey auctions. I amindebted to professor de Frutos for pointing that to me.17hich is consistent with this assumption. However, in many cases the sellers can-not observe the trade between other parties. Moreover, government regulationsmight forbid the use of such information in contracts. For these reasons, I considerhere a variation of the game, named private common agency.In this modified version of the game, the agent’s action a is a vector of dimen-sion ( n ) , which is equal to the number of principals. Formally, a = ( a , a , ..., a i , ..., a n ) ∈ A = × A i ⊂ R n . Moreover, the bidding functions of the principals take the form: b i : A i → R such that b i = b i ( a i )and the utility function of principal i is: u i : A i × R → R such that u i = u i ( a i , b i ) . Finally, the two functions that define feasibility are: b i : A i → R and b i : A i → R. The rest of the setting is as in section 2.I call the game defined above a private game . In this new setting the defini-tions of all other concepts that appear in section 2 must be modified appropriately.These modifications are straightforward, so I omit them and jump directly to themain result.
Proposition 6.
Validity in private games
All truthful equilibria in private games that exhibit opposing monotonicity arevalid.Proof: See appendix B.1.4Chiesa and Denicol`o (2009), also study private games and find that all equilib-ria are efficient and that truthful equilibria satisfy proposition 3A. Their settingdiffers from mine in two ways. First, Chiesa and Denicol`o (2009) assume that theprincipals also choose a subset of A i on which they condition their bids. Second,in their model, these authors allow only for quasi linear utility functions.I turn now to a further discussion of efficiency.18 Discussion
In this section I discuss games with externalities that do not satisfy propositions2-4. Specifically, I investigate the existence of efficient equilibria in such games.As it turns out, efficient equilibria might or might not exist depending on thespecifics of the model. I discuss this issue with the help of two examples. Theseexamples also clarify the relationship between the paper in hand and the existingliterature on common agency with externalities. I start with an example in whichthere are no efficient equilibria.
Example 1
Assume there are two principals ( i = 1 ,
2) with utility functions u i = a − b i + γb j ,in which a ∈ [0 ,
1] is the agent’s action, b i ∈ [0 , a ] is the bid of principal i and γ isa positive parameter. The utility of the agent is u = b + b . First, I assume γ = 2 . In this case, the example satisfies opposing monotonicity,but violates small externalities. Moreover, as I show in appendix B.2.1 there existsonly one allocation, which is both symmetric and efficient. In this allocation: a = 1 , b = b = 1 and u = u = u = 2 . Henceforth, I call this allocation,allocation A. However, allocation A is not an equilibrium. To see this, consider anybidding functions such that b i (1) = 1 for both i and think of the following deviationby principal 1: b = 0 , a = 1 and b = 0 for all a (cid:54) = 1 . Then, no matter what isthe bidding function of principal 2, the agent chooses a = 1 and total bids decrease.Moreover, as I also show in appendix B.2.1, there are no efficient equilibria in thisexample. On the contrary, there is a unique symmetric inefficient equilibriumwhich is intuitive . Specifically, consider the following allocation, which I nameallocation B: a = 1 , b = b = 0 . This allocation yields u = 0 , u = u = 1 andcan be supported as an equilibrium, by the constant bidding functions b i ( a ) = 0for all a ∈ [0 , . Allocations A and B illustrate the cause of efficiency failure in common agencywith externalities. This cause is a prisoners’ dilemma. Specifically, the princi- Also, there exist many asymmetric inefficient equilibria that are supportedby implausible off equilibrium strategies. I provide an example in appendixB.2.1. 19als can benefit from committing to high bids, as in allocation A. However, thiscommitment is not viable. This is so, because each principal has the motive tounilaterally deviate from any such “agreement” and offer smaller bids. This de-viation generates a race to the bottom which leads to the inefficient equilibriumB. This prisoners’ dilemma also characterizes the examples provided by Peters(2001) and Martimort and Stole (2002). These authors, observe that externalitiesin common agency models might lead to inefficient equilibria. Following this ob-servation, they suggest an extension of the common agency model. This extension,allows bids to depend on the biding functions of the other principal and restoresefficiency. This idea in terms of example 1 is as follows: Allocation A is not anequilibrium because principal 1 deviates. However, if the bids of one principaldepend on the biding function of the other, then principal 2 can use his biddingfunction to punish the deviating behaviour and support the equilibrium.In contrast to this approach, I bypass the solution of the efficiency issue al-together. Instead, I notice that in certain cases a prisoners’ dilemma does notappear. Indeed, if γ = 0 . , example 1 satisfies proposition 5. Thus, if a truthfulequilibrium exists it is efficient. Specifically, allocation B is such an equilibrium.This result, follows from the fact that if γ = 0 . , the principals can not benefitfrom high bids. In general, small externalities do not allow for prisoner dilemmasand therefore lead to efficient equilibria. A similar argument holds for symmetricnegative externalities.Now I turn to an example which violates proposition 5, but nevertheless hasan efficient equilibrium. Example 2
Consider a variation of example 1 in which the utility of the agent is u = − ( b + b )and γ = 2 . This example violates both opposing monotonicity and small external-ities. Yet, allocation B can still be supported as an efficient equilibrium, by theconstant bidding functions b i ( a ) = 0 for all a. In this case, allocation B is efficient because the agent dislikes bids. In anyother allocation with a = 1 and positive bids, the principals might be better off20ut the position of the agent deteriorates. This is in contrast to example 1, inwhich both principals and agent get worse off, as a result of a decrease in bids.In example 2, the prisoners’ dilemma between principals remains, however it doesnot hinder efficiency, due to the characteristics of the agent’s utility function.Yet, although allocation B is an efficient equilibrium it is not also a truthfulone. This is so, because the structure of truthful bidding functions renders themmeaningless without opposing monotonicity. Indeed, in such cases truthful biddingfunctions fail, since they imply that the principals offer to the agent something hedislikes. Boultzis (2015), considers an example of this situation.Examples 1 and 2 above discuss the role of small externalities and opposingmonotonicity in the efficiency of equilibria. In appendix B.2.2, I provide someadditional examples that outline the role of symmetry and quasi-concavity.Next, I consider economic applications. In this section, I provide two economic applications of proposition 5.
This is a case of market monotonicity and no externalities.Assume n = 2. The two principals are sellers who sell a homogeneous good.The agent is a buyer who has decided to buy q units of the good. For example,think of a government which is in the market for a fixed number of warships.First, the two sellers submit an offer for a unit price that depends on thequantity that the buyer buys from each seller. Then, the buyer decides how muchto buy from each seller. This model is known in the literature as split-awardprocurement . Although in this model each seller conditions his price exclusivelyon the quantity he sells to the buyer it is still a public common agency model.This is so because the buyer’s choice is essentially one dimensional. In particular See Anton and Yao (1989) and Chiesa and Denicol`o (2009).21hen the agent chooses to buy quantity q from principal 1, this decision alsodetermines the quantity he buys from principal 2 ( q = q − q ) . The profit function of seller i is:Π i = p i q i − cq i . Here, p i is the unit price, q i stands for quantity and c > q between the two sellers in a way that minimizeshis expenditure. Thus, the utility of the buyer is: u = − p q − p ( q − q ) . The reservation price for the buyer is p. You can think of the reservation price asthe cost of not buying a unit or as the unit price offered by an outside source.The bounds that define feasibility for prices are : p i ( q i ) = cq i ≥ p min and p i ( q i ) = p. These conditions reflect the fact that prices must be positive, must not be lessthan the average cost and must not exceed the reservation price. Furthermore,assume p > cq. This model satisfies market monotonicity and lacks externalities. Thereforeit satisfies proposition 5. As I show in appendix B.3.1, solving for a truthfulequilibrium yields the following symmetric outcome: q i = q , p i = 3 cq , Π i = cq p i ( q i ) = cq q i + cq i if q i ≥ p − √ p − c q c p if q i < p − √ p − c q c In this model, total cost decreases with the number of producers, because ofthe increasing marginal cost. Specifically, the total cost of producing quantity q of the good is cq if there is only one producer, while it is cut in half, if there aretwo producers. Consequently, a familiar result emerges.22pecifically the profits of each principal equal the decrease in the total cost dueto his participation in production. This result is standard for truthful equilibriain common agency games. In this respect, Bergemann and V¨alim¨aki (2003) showthat in such cases, each principal receives his contribution to the social surplus .I turn now to an application with externalities. I consider an example with lobbying monotonicity and symmetric negative exter-nalities. In particular, I introduce lobbying to a local public goods model inspiredby Persson and Tabellini (1994).Consider a federal country with n identical states, each populated by oneindividual. In this country, there is a federal government with the sole purpose toredistribute income across states. This government imposes a tax or subsidy t i toeach of the states. These taxes satisfy: (cid:88) i t i = 0 . Each state offers to the federal government a bribe b i in order to affect the choiceof t i . In this respect the utility of the federal government is given by: u = (cid:88) i b i . Moreover, there are two goods in the economy. These goods are the privategood c and the public good G. Each state finances the public good by offering avoluntary contribution g i . Thus the total amount of public good equals: G = (cid:88) i g i . An example of such a public good is public safety. Each state decides indepen-dently how much to spend on the security of its airport. However, since terrorists On the structure of payoffs in such games see also Laussel and Le Breton(2001) and Villemeur and Versaevel (2003).23ho arrive in one state can move freely to all states, this spending affects safetyin the whole country.Total spending in each state is financed by an endowment e. This endowmentplus or minus the federal transfer t i is used for private good consumption, contri-bution to the public good and bribes. Therefore, the budget constraint for eachstate is: e + t i = c i + b i + g i , where t i ∈ [ − e, e ] . Finally the utility in state i is : u i = c i G. The timing in this model is as follows: First, a common agency game deter-mines bribes and transfers. Then, the two states decide on their public goodcontributions simultaneously .Following this timing, I start by determining the equilibrium public good con-tributions. Define, e i = e + t i − b i . Then, the unique Nash equilibrium yields g i = e i − (cid:80) i e i n +1 and G = (cid:80) i e i n +1 , which implies that: u i = (cid:18) (cid:80) i e i n + 1 (cid:19) This function is obviously concave and symmetric with respect to b i , b j . Moreover, b i is appropriately bounded since:0 ≤ b i ≤ e + t i ≤ e. This chain of inequalities implies that b i ( · ) = b j ( · ) and b i ( · ) = b j ( · ) . Moreover, the derivatives of interest are as follows: ∂u ∂b i =1 > ∂u i ∂b i = − G < ∂u i ∂ ˜ b − i = − G < Bergemann and V¨alim¨aki (2003) and Bhaskar and To (2004) also discuss multistage models with a separate common agency stage.24hus, this game satisfies lobbying monotonicity and negative externalities.Furthermore it is symmetric and quasi concave and therefore exhibits symmetricnegative externalities. Hence, truthful equilibria are valid. Solving for a truthfulequilibrium yields: b = b = 0 and t i ∈ [ − e, e ] s.t. (cid:88) i t i = 0 . Specifically, the functions b i = 0 for all feasible t i can be supported as truthfulresponses to each other, relative to the equilibrium utility level u i = (cid:0) nen +1 (cid:1) . Moreover, these functions trivially satisfy conditions (Ai) and (Aii). Finally, whenbribes equal their maximum value ( e + t i ) the utility of the principals alwaysequals zero which is its lower bound. Thus, this application satisfies strong deeppockets, which implies that the suggested allocations also satisfy condition (Bii)and therefore constitute truthful equilibria.In this model the agent has no “bargaining power”. Thus, as in similar modelswithout externalities he ends with nothing. Specifically, since the utility of eachstate depends on total income, the government can not affect it by redistributivetransfers. Thus, there is no room for bribes. The two applications above belong in two of the general settings identified byproposition 5. I consider them here because they are simple and characteristicof how proposition 5 can be used. However, proposition 5 considers two moresettings, which I briefly review in the remaining of this subsection.The first setting incorporates applications which exhibit lobbying monotonicityand no externalities. This is the case discussed by Dixit et al. (1997). This typeof models is often used to describe situations associated with lobbying. I donot discuss them any further, since they have been studied extensively in theliterature . Early examples of this literature are Grossman and Helpman (1994), Dixit(1996), Persson (1998), Mitra (1999) e.t.c. More recent examples are Cam-25he second setting includes models with lobbying monotonicity and small ex-ternalities. Such situations occur in lobbying models, in which the relation betweenthe principals goes beyond sharing a common agent, very much like in the appli-cation in 6.2 above. For example, consider a variation of this application in whichthe states have different endowments and the good G is not a pure public good.In this respect, assume that G i = g i + γ (cid:80) i (cid:54) = j g j , where G i is the quantity of good G consumed by state i and γ ∈ (0 ,
1) is a parameter capturing the externality be-tween states. This modified application exhibits lobbying monotonicity and smallexternalities .I turn now to the concluding remarks. In this paper I provide a set of conditions under which truthful equilibria arevalid in common agency models. These conditions generalize the work of Dixitet al. (1997) and Bernheim and Whinston (1986) on this issue. Furthermore, Iidentify two new families of economic applications to which these conditions apply.In this regard, this paper shows that the scope of truthful equilibria is broaderthan believed so far.In terms of future research my results can be useful in two ways. First, theconditions listed in proposition 5 apply in a wide variety of economic models. Thevalidity of truthful equilibria in these models provides for a simple and intuitiveway of solving an otherwise difficult problem. Second, propositions 2-4 can helpidentify additional settings in which truthful equilibria are valid in the future.Thus, the scope of truthful equilibria might be extended even further.pante and Ferreira (2007), Aidt and Hwang (2008) and Esteller-Mor´e etal. (2012) among many others. For a related work see also Felli and Merlo(2006). Additional details are available upon request26
Appendix
Proof of proposition 4
Assume the contrary is true. Then, there exists a feasible pair ( a ∗ , b ∗ ) and atruthful equilibrium of the game ( a o , b o ( · )) , such that u ( a ∗ , b ∗ ) ≥ u ( a o , b o ( a o ))and u ( a ∗ , b ∗ ) ≥ u ( a o , b o ( a o )) , with at least one strict inequality. These inequalitiesyield: u ( a ∗ , b ∗ ) ≥ u ( a o , b o ( a o )) ≥ u ( a ∗ , b o ( a ∗ ))and u ( a ∗ , b ∗ ) ≥ u ( a o , b o ( a o )) ≥ u ( a ∗ , b o ( a ∗ )) . The last inequality in the first expression above holds because ( a o , b o ( · )) is anequilibrium. Also, the last inequality in the second expression is due to deeppockets.The two chains of inequalities above imply that u ( a ∗ , b ∗ ) ≥ u ( a ∗ , b o ( a ∗ ))and u ( a ∗ , b ∗ ) ≥ u ( a ∗ , b o ( a ∗ )) , with at least one strict inequality. Then, there isa contradiction. Indeed, if u ( a ∗ , b ∗ ) > u ( a ∗ , b o ( a ∗ )) it follows from conflict ofinterests that there is an i, such that u i ( a ∗ , b ∗ ) > u i ( a ∗ , b o ( a ∗ )) , which contradictsthe second chain of inequalities. Alternatively, if u ( a ∗ , b ∗ ) > u ( a ∗ , b o ( a ∗ )) conflictof interests implies that u ( a ∗ , b ∗ ) > u ( a ∗ , b o ( a ∗ )) , which contradicts the first chainof inequalities. Q.E.D. References
Aidt, Toke S, and Uk Hwang. 2008. “On the internalization of cross-national ex-ternalities through political markets: the case of labour standards”.
Journal ofInstitutional and Theoretical Economics JITE
164 (3): 509–533.Anton, James J, and Dennis A Yao. 1989. “Split awards, procurement, and inno-vation”.
RAND Journal of Economics
20 (4): 538–552.Bergemann, Dirk, and Juuso V¨alim¨aki. 2003. “Dynamic common agency”.
Journalof Economic Theory
111 (1): 23–48.27ernheim, B Douglas, and Michael D Whinston. 1986. “Menu auctions, resourceallocation, and economic influence”.
Quarterly Journal of Economics
101 (1):1–31.Bhaskar, Venkataraman, and Ted To. 2004. “Is perfect price discrimination reallyefficient? An analysis of free entry”.
RAND Journal of Economics
Economics Letters
Journal of Public Economics
91 (5-6): 993–1021.Chiesa, Gabriella, and Vincenzo Denicol`o. 2009. “Trading with a common agentunder complete information: A characterization of Nash equilibria”.
Journalof Economic Theory
144 (1): 296–311.Dasgupta, Partha, and Eric Maskin. 2000. “Efficient auctions”.
Quarterly Journalof Economics
115 (2): 341–388.Dixit, Avinash. 1996. “Special-interest lobbying and endogenous commodity tax-ation”.
Eastern Economic Journal
22 (4): 375–388.Dixit, Avinash, Gene M Grossman, and Elhanan Helpman. 1997. “Common agencyand coordination: General theory and application to government policy mak-ing”.
Journal of Political Economy
105 (4): 752–769.Dixit, Avinash, Gene Grossman, and Elhanan Helpman. 1996.
Common Agencyand Coordination: General Theory and Application to Tax Policy . Tech. rep.CEPR Discussion Papers.Esteller-Mor´e, Alejandro, Umberto Galmarini, and Leonzio Rizzo. 2012. “Verticaltax competition and consumption externalities in a federation with lobbying”.
Journal of Public Economics
96 (3-4): 295–305.Felli, Leonardo, and Antonio Merlo. 2006. “Endogenous lobbying”.
Journal of theEuropean Economic Association
Journal of Economic Theory
AmericanEconomic Review
84 (4): 833.Kirchsteiger, Georg, and Andrea Prat. 2001. “Inefficient equilibria in lobbying”.
Journal of Public Economics
82 (3): 349–375.Ko, Chiu Yu. 2017. “A note on budget constraints and outside options in commonagency”.
Theory and Decision
83 (1): 95–106.— . 2011. “Menu auctions with non transferable utilities and budget constraints”.
Boston College Department of Economics: Working Paper No 787 .Laussel, Didier, and Michel Le Breton. 2001. “Conflict and cooperation: The struc-ture of equilibrium payoffs in common agency”.
Journal of Economic Theory
100 (1): 93–128.Martimort, David, and Lars Stole. 2003. “Contractual Externalities and CommonAgency Equilibria”.
BE Journal of Theoretical Economics
RAND journal of economics
40 (1): 78–102.— . 2009b. “Selecting equilibria in common agency games”.
Journal of EconomicTheory
144 (2): 604–634.— . 2002. “The revelation and delegation principles in common agency games”.
Econometrica
70 (4): 1659–1673.Mitra, Devashish. 1999. “Endogenous lobby formation and endogenous protection:a long-run model of trade policy determination”.
American Economic Review
89 (5): 1116–1134.Persson, Torsten. 1998. “Economic policy and special interest politics”.
EconomicJournal
108 (447): 310–327.Persson, Torsten, and Guido Tabellini. 1994. “Does centralization increase the sizeof government?”
European Economic Review
38 (3-4): 765–773.29eters, Michael. 2001. “Common agency and the revelation principle”.
Economet-rica
69 (5): 1349–1372.Peters, Michael, and Bal´azs Szentes. 2012. “Definable and contractible contracts”.
Econometrica
80 (1): 363–411.Prat, Andrea, and Aldo Rustichini. 2003. “Games played through agents”.
Econo-metrica
71 (4): 989–1026.Segal, Ilya, and Michael D Whinston. 2003. “Robust predictions for bilateral con-tracting with externalities”.
Econometrica
71 (3): 757–791.Szentes, Bal´azs. 2014. “Contractible contracts in common agency problems”.
Re-view of Economic Studies
82 (1): 391–422.Villemeur, Etienne Billette de, and Bruno Versaevel. 2003. “From private to publiccommon agency”.
Journal of Economic Theory
111 (2): 305–309.30
Online Appendix
B.1 Proofs
B.1.1 Proof of proposition 1Proof of (i)
I show that the first version of strong deep pockets (D.2.1) im-plies weak deep pockets (D.1). Let ( a o , b o ( · )) be a truthful equilibrium and u o = u ( a o , b o ( a o )) . Also, let ( a ∗ , b ∗ ) be a feasible pair. Because of strong deeppockets, u oi ≥ u i ( a, b i ( a ) , b o − i ( a )) for all i and a ∈ A. Then,the definition of truthfulbidding functions implies either a) b oi ( a ∗ ) = φ i ( a ∗ ; u o , b o − i ( · )) or b) b oi ( a ∗ ) = b i ( a ∗ ) . If (a) holds, then u i ( a ∗ , b o ( a ∗ )) = u oi , while if (b) holds, then u i ( a ∗ , b o ( a ∗ )) < u oi . Taking (a) and (b) together yields u oi = u i ( a o , b o ( a o )) ≥ u i ( a ∗ , b o ( a ∗ )) , for all i. Since this inequality holds for all feasible pairs, it also holds for the feasible pair inthe definition of weak deep pockets. The proof that the second version of strongdeep pockets implies weak deep pockets is essentially the same. Q.E.D.
Proof of (ii)
I consider market monotonicity. The proof for lobbying monotonic-ity is essentially the same. The fact that market monotonicity implies conflict ofinterests in models without externalities is obvious. In what follows I prove thatit also implies weak deep pockets.Let ( a o , b o ( · )) be a truthful equilibrium and u o = u ( a o , b o ( a o )) . Then, follow-ing the definition of weak deep pockets consider a feasible pair ( a ∗ , b ∗ ) such that u ( a ∗ , b ∗ ) ≥ u ( a o , b o ( a o )) = u ( a ∗ , φ ( a ∗ ; u o )) . Because u ( · ) is increasing in own bidsit must be that b ∗ i ≥ φ i ( a ∗ ; u o ) for all i . Furthermore, since b ∗ i is feasible itfollows that b max ≥ b ∗ i ≥ b i ( a ∗ ). Therefore, b max ≥ φ i ( a ∗ ; u o ) . This last inequal-ity and the definition of truthful responses imply that either b oi ( a ∗ ) = φ i ( a ∗ ; u o )or b oi ( a ∗ ) = b i ( a ∗ ) . In turn, these equalities along with the feasibility of b ∗ i im-ply that b ∗ i ≥ b oi ( a ∗ ) for all i. Then, the fact that u ( · ) is decreasing in all bidsyields: u ( a ∗ , b o ( a ∗ )) ≥ u ( a ∗ , b ∗ ) ≥ u ( a o , b o ( a o ) . If any of these two inequal-ities is strict then we have a contradiction since ( a o , b o ( · )) is an equilibrium.Thus, u ( a ∗ , b o ( a ∗ )) = u ( a ∗ , b ∗ ) = u ( a o , b o ( a o )) . Now, if there exists an i suchthat u i ( a ∗ , b oi ( a ∗ )) > u oi we arrive at a contradiction since the pair ( a o , b o ( · )) vio-31ates trivially the definition of equilibrium. Indeed, in such a case it follows from u ( a ∗ , b o ( a ∗ )) = u ( a o , b o ( a o ) that a ∗ ∈ arg max a ∈ A u ( a, b oi ( a ) , b o − i ( a )) . Thus, a o cannot be part of an equilibrium. Therefore, u ( a o , b o ( a o )) ≥ u ( a ∗ , b o ( a ∗ )) . Q.E.D.
Proof of (iii) Part 1.
First, I show that lobbying monotonicity and small externalities imply conflict ofinterests.The total differential of the utility of the principals is du i = ∂u i ∂b i db i + ∂u i ∂ ˜ b − i d ˜ b − i . Rearranging terms yields: du i = ( ∂u i ∂b i − ∂u i ∂ ˜ b − i ) db i + ∂u i ∂ ˜ b − i d ˜ b. In order to proceed I need to show two things. First, that when the agent’sutility increases ( d ˜ b > du i < i ).I start with negative externalities ( ∂u i ∂ ˜ b − i < . In this case small externalitiesimply that ∂u i ∂b i − ∂u i ∂ ˜ b − i < . Then, since d ˜ b > i such that db i > . As a result, du i < ∂u i ∂ ˜ b − i > . Consider the principal withthe largest increase in bids. Then, for this principal ndb i ≥ d ˜ b. Substituting thisexpression in the expression for du i yields du i ≤ ( ∂u i ∂b i + ( n − ∂u i ∂ ˜ b − i ) db i < . Thelast inequality follows from small externalities, since ∂u i ∂b i + ( n − ∂u i ∂ ˜ b − i < . The second thing I need to show is that if all principals are weakly betteroff and at least one is strictly better off ( du > d ˜ b < . Assume the contrary, d ˜ b ≥ . Then, if db i = 0 for all i it mustbe that du i = 0 for all i , which is a contradiction. Thus, for at least an i , it mustbe that db i >
0. Then, for this principal the analysis above implies that du i < Part 2.
In this part, I show that small externalities along with lobbying monotonicity im-ply weak deep pockets. I start with the proof of a lemma that I will use laterduring the rest of the proof.
Lemma 1 b, b ´ are feasible for some a ∈ A and there exists an i , such that b i ´ > b i and u i ( a, b i ´ , ˜ b ´ − i ) ≥ u i ( a, b i , ˜ b − i ) then, ˜ b ´ < ˜ b. Proof of lemma 1:Assume the contrary: ˜ b ´ ≥ ˜ b ⇒ b i ´ − b i ≥ ˜ b − i − ˜ b ´ − i > . The last inequality fol-lows from b i ´ > b i , u i ( a, b i ´ , ˜ b ´ − i ) > u i ( a, b i , ˜ b − i ) and negative externalities. Consider κ ≥ u i ( κ ) = u i ( a, b i + κ, ˜ b − i − κ ) . Then, ∂u i ∂κ = ∂u i ∂b i − ∂u i ∂ ˜ b − i < . Thelast inequality is due to small negative externalities. Then, using ∂u i ∂κ < u i (0) > u i ( b i ´ − b i ) or u i ( a, b i , ˜ b − i ) > u i ( a, b i ´ , ˜ b − b i ´) ≥ u i ( a, b i ´ , ˜ b ´ − i ) . The last in-equality is because by assumption ˜ b ´ ≥ ˜ b and negative externalities. This last chainof inequalities implies u i ( a, b i , ˜ b − i ) > u i ( a, b i ´ , ˜ b ´ − i ) , which is a contradiction. Thisconcludes the proof of the lemma.I proceed now with the rest of the proof of part 2.The properties of ( a ∗ , b ∗ ) and ( a o , b o ( a o )) in the definition of weak deep pocketsimply that u ( a ∗ , ˜ b ∗ ) ≥ u ( a o , ˜ b o ( a o )) ≥ u ( a ∗ , ˜ b o ( a ∗ )) . In turn, this chain of in-equalities implies that ˜ b ∗ ≥ ˜ b o ( a ∗ ) . Assume there is an i , such that u i ( a ∗ , b o ( a ∗ )) >u i ( a o , b o ( a o )) . Then, if ˜ b ∗ = ˜ b o ( a ∗ ) it follows that u ( a ∗ , ˜ b ∗ ) = u ( a o , ˜ b o ( a o )) = u ( a ∗ , ˜ b o ( a ∗ )) . In this case there is a contradiction since the pair ( a o , b o ( · )) violatestrivially the definition of equilibrium. Specifically, it follows from u ( a ∗ , ˜ b o ( a ∗ )) = u ( a o , ˜ b o ( a o )) that a ∗ ∈ arg max a ∈ A u ( a, ˜ b o ( a )) . This last observation, along with u i ( a ∗ , b o ( a ∗ )) > u i ( a o , b o ( a o )) imply that a o can not be part of an equilibrium.Therefore, ˜ b ∗ > ˜ b o ( a ∗ ) . Since the utility of the agent is strictly increasing in total bids, u ( a ∗ , ˜ b ∗ ) >u ( a ∗ , ˜ b o ( a ∗ )) . Then, following part 1 of this proof, there exists an i such that u i ( a ∗ , b o ( a ∗ )) > u i ( a ∗ , b ∗ ) ≥ u i ( a o , b o ( a o )) . For this i , the definition of truthfulresponses implies that b oi ( a ∗ ) = b i ( a ∗ ) ≥ b ∗ i . I consider two cases. First, I consider positive externalities. Since u i ( a ∗ , b o ( a ∗ )) >u i ( a ∗ , b ∗ ) and b oi ( a ∗ ) ≥ b ∗ i , it must also be that ˜ b o − i ( a ∗ ) ≥ ˜ b ∗− i with at least one strictinequality. However, in this case ˜ b o ( a ∗ ) > ˜ b ∗ , which is a contradiction.33econd, I turn to negative externalities. In this case, I start by observing that b oi ( a ∗ ) ≥ b ∗ i does not hold for all i, because if it does, it follows that ˜ b o ( a ∗ ) ≥ ˜ b ∗ , which is a contradiction. Thus, there is at least an i, such that b oi ( a ∗ ) ˜ b ∗ , which is acontradiction. Thus u ( a o , b o ( a o )) ≥ u ( a ∗ , b o ( a ∗ )) . Q.E.D.
Proof of (iv)Conflict of interests.
I prove first that lobbying monotonicity and symmetric negative externalities im-ply conflict of interests at symmetric allocations.Consider a game that exhibits lobbying monotonicity and symmetric negativeexternalities and let ( a, b ∗ ) , ( a, b ) be two feasible pairs and also let ( a, b ) be sym-metric. Then I will prove that, (i) if ˜ b ∗ > ˜ b there is an i such that u i ( a, b ∗ ) < u i ( a, b )and (ii) if u ( a, b ∗ ) > u ( a, b ) then ˜ b ∗ < ˜ b. I start with the proof of (i). I assume the contrary: u i ( a, b ∗ ) ≥ u i ( a, b ) , for all i. Then, because ˜ b ∗ > ˜ b, there exists ξ i ∈ [ − b i , b max − b i ] for all principals i, suchthat b ∗ i = b i + ξ i and (cid:80) i ξ i > . Moreover, because of the symmetry of the gameand ( a, b ), it follows from u i ( a, b ∗ ) ≥ u i ( a, b ) that u k ( a, b ∗ i , ˜ b ∗− i ) ≥ u k ( a, b k , ˜ b − k ) forall i and k. Then, the quasi-concavity of u k ( · ) yields u k ( a, n (cid:80) i b ∗ i , n (cid:80) i ˜ b ∗− i ) ≥ u k ( a, b k , ˜ b − k ) . However, n (cid:80) i b ∗ i = n (cid:80) i b i + n (cid:80) i ξ i = b k + n (cid:80) i ξ i > b k . Also, n (cid:80) i ˜ b ∗− i = n − n (cid:80) i b i + n − n (cid:80) i ξ i = ˜ b − k + n − n (cid:80) i ξ i > ˜ b − k . Then, because of nega-tive externalities u k ( a, n (cid:80) i b ∗ i , n (cid:80) i ˜ b ∗− i ) < u k ( a, b k , ˜ b − k ) , which is a contradiction.I turn now to (ii). Specifically, I must show that if u ( a, b ∗ ) > u ( a, b ) then˜ b ∗ < ˜ b. Assume the contrary, ˜ b ∗ ≥ ˜ b. If ˜ b ∗ > ˜ b then the analysis above yields acontradiction. Therefore, ˜ b ∗ = ˜ b. Using symmetry, u ( a, b ∗ ) > u ( a, b ) implies that u k ( a, b ∗ i , ˜ b ∗− i ) ≥ u k ( a, b k , ˜ b − k ) for all i and k, with at least one strict inequality.Then, the quasi-concavity of u k ( · ) yields u k ( a, n (cid:80) i b ∗ i , n (cid:80) i ˜ b ∗− i ) > u k ( a, b k , ˜ b − k ) . However, n (cid:80) i b ∗ i = ˜ b ∗ n = ˜ bn = b k . Also, n (cid:80) i b ∗− i = ( n − b ∗ n = ( n − bn = ( n − b k =˜ b − k . These equalities yield u k ( a, n (cid:80) i b ∗ i , n (cid:80) i ˜ b ∗− i ) = u k ( a, b k , ˜ b − k ) , which is a con-34radiction. Weak deep pockets
I continue to prove that symmetric negative externalities and lobbying mono-tonicity imply weak deep pockets. In this regard consider such a game and let( a o , b o ( · )) be a symmetric truthful equilibrium and ( a ∗ , b ∗ ) be a feasible pair suchthat u ( a ∗ , b ∗ ) ≥ u ( a o , b o ( a o )) and u ( a ∗ , b ∗ ) ≥ u ( a o , b o ( a o )) with at least one strictinequality. Then, I will show that u ( a o , b o ( a o )) ≥ u ( a ∗ , b o ( a ∗ )) . I start by assuming the opposite. That is, there exists an i for which u i ( a ∗ , b o ( a ∗ )) >u i ( a o , b o ( a o )) . Consider first the case u ( a ∗ , b ∗ ) > u ( a o , b o ( a o )) . Then, u ( a ∗ , b ∗ ) > u ( a o , b o ( a o )) ≥ u ( a ∗ , b o ( a ∗ )) implies that u ( a ∗ , b ∗ ) > u ( a ∗ , b o ( a ∗ )) . Moreover, for the princi-pal i above, the definition of truthful responses yields b oi ( a ∗ ) = b i ( a ∗ ) . Then,symmetry implies that b oi ( a ∗ ) = b i ( a ∗ ) for all i. Therefore, ˜ b o ( a ∗ ) ≥ ˜ b ∗ and u ( a ∗ , b o ( a ∗ )) ≥ u ( a ∗ , b ∗ ) which is a contradiction. The last inequality followsfrom the fact that the game is cumulative.Now I turn to the case u ( a ∗ , b ∗ ) = u ( a o , b o ( a o )) . In this case, if u ( a o , b o ( a o )) >u ( a ∗ , b o ( a ∗ )) the analysis above can be repeated. Thus, u ( a ∗ , b ∗ ) = u ( a o , b o ( a o )) = u ( a ∗ , b o ( a ∗ )) . This equality implies that a ∗ also maximizes the utility of the agentwhen the principals submit b o ( · ). Moreover, because of assuming the contrary inthe beginning of the proof, there exists an i such that u i ( a ∗ , b o ( a ∗ )) > u i ( a o , b o ( a o )) . The existence of this principal leads to a contradiction, since a o trivially violatesthe definition of equilibrium. Q.E.D. B.1.2 Proof of proposition 2Proof of proposition 2
I prove proposition 2 under the assumption of market monotonicity. The prooffor lobbying monotonicity is very similar.I will show that if any best response of a principal yields a certain utility levelthen the truthful response relative to this utility level is also a best response. Inthis respect, let b oi ( · ) be a best response of principal i to the bidding functions35 o − i ( · ) of the other principals. Then, there exists a o ∈ arg max a ∈ A u ( a, b o ( a )), suchthat there does not exist a feasible pair ( a ∗ , b ∗ i ( · )), such that u i ( a ∗ , b ∗ i ( a ∗ ) , b o − i ( a ∗ )) >u i ( a o , b o ( a o )) and a ∗ ∈ arg max a ∈ A u ( a, b ∗ i ( a ) , b o − i ( a )) .Define, u oi = u i ( a o , b o ( a o )) and the truthful response of principal i to the bid-ding functions of the other principals relative to u oi as b Ti ( a ; b o − i ( · ) , u oi ). For theshake of simplicity in the remaining of the proof I suppress the other argumentsand write b Ti ( a ) . Finally, I define the set A ´= arg max a ∈ A u ( a, b Ti ( a ) , b o − i ( a )) . If a o ∈ A ´then b Ti ( a ) is trivially a best response since b Ti ( a o ) = b oi ( a o ) . I turn now to the case in which a o / ∈ A ´ . Consider any a ´ ∈ A ´ . Then, I claim that u ( a ´ , b Ti ( a ´) , b o − i ( a ´)) > u ( a ´ , b o ( a ´)) . If not, then u ( a ´ , b o ( a ´)) ≥ u ( a ´ , b Ti ( a ´) , b o − i ( a ´)) >u ( a o , b Ti ( a o ) , b o − i ( a o )) = u ( a o , b o ( a o )) . The strict inequality follows from the factthat a o / ∈ A ´ while a ´ ∈ A ´ and the equality from the definition of b Ti ( · ) . Thus, u ( a ´ , b o ( a ´)) > u ( a o , b o ( a o )) which is a contradiction because a o ∈ arg max a ∈ A u ( a, b o ( a )) . Furthermore, because of market monotonicity the utility of the agent is de-creasing in all bids. Therefore, u ( a ´ , b Ti ( a ´) , b o − i ( a ´)) > u ( a ´ , b o ( a ´)) implies that b Ti ( a ´) < b oi ( a ´) ≤ b i ( a ´) . Then, following the definition of truthful responses ei-ther b Ti ( a ´) = φ i ( a ´; u oi , b − i ( · )) or b Ti ( a ´) = b i ( a ´) . The last result combined with theincreasing utility of the principals in own bids and the definition of truthful re-sponses yields u i ( a ´ , b Ti ( a ´) , b o − i ( a ´)) ≥ u oi which proves that b Ti ( · ) is a best response.Q.E.D. B.1.3 Proof of proposition 3
I prove proposition 3 under the assumption of market monotonicity. The prooffor lobbying monotonicity is very similar.
Proof of 3A
I start with the following lemma.
Lemma
A feasible pair ( a o , b o ( · )) is an equilibrium if and only if:(i) a o ∈ arg max a ∈ A u ( a, b o ( a )) 36ii)For all i , ( a o , b oi ( a o )) ∈ arg max ( a,b i ) u i ( a, b i , b o − i ( a )) subject to a ∈ A , b i = b i ( a ) forsome feasible bidding function b i ( · ) and u ( a, b i , b o − i ( a )) ≥ max a ´ ∈ A u ( a ´ , b max , b o − i ( a ´)) . Proof of the Lemma:Necessity:Assume that ( a o , b o ( · )) is an equilibrium but it does not solve the maximiza-tion problem in condition (ii) of the lemma. Then, there exists an i and afeasible pair ( a ∗ , b ∗ i ) which satisfies the constraints in condition (ii) and yields u i ( a ∗ , b ∗ i , b o − i ( a ∗ )) > u i ( a o , b o ( a o )) . In this case though, I can show that ( a o , b o ( · )) is not an equilibrium whichis a contradiction. In order to show this contradiction I need to prove thatthere exists a feasible bidding function b ∗ i ( · ) such that b ∗ i ( a ∗ ) = b ∗ i and a ∗ ∈ arg max a ∈ A u ( a, b ∗ i ( a ) , b o − i ( a )) . Since ( a ∗ , b ∗ i ) satisfies the constraints in condition (ii), there exists a feasiblebidding function ˆ b i ( · ) such that ˆ b i ( a ∗ ) = b ∗ i . Define the function ϕ i : A → R implicitly, through u ( a, ϕ i , b o − i ( a )) = u ( a ∗ , b ∗ i , b o − i ( a ∗ )) . The function ϕ i ( · ) alwaysexists because the utility of the agent is strictly decreasing in all bids. Furthermore, ϕ i ( a ∗ ) = b ∗ i . Moreover, because of condition (ii), u ( a ∗ , b ∗ i , b o − i ( a ∗ )) ≥ max a ´ ∈ A u ( a ´ , b max , b o − i ( a ´))and therefore u ( a, ϕ i ( a ) , b o − i ( a )) = u ( a ∗ , b ∗ i , b o − i ( a ∗ )) ≥ max a ´ ∈ A u ( a ´ , b max , b o − i ( a ´)) ≥ u ( a, b max , b o − i ( a )) for all a ∈ A. Since the agent’s utility is strictly decreasing in all bids, this chain of inequal-ities implies that ϕ i ( a ) ≤ b max , for all a ∈ A. Define b ∗ i ( a ) = max [ ϕ i ( a ) , ˆ b i ( a )] . Due to ϕ i ( a ) ≤ b max and the fact that ˆ b i ( · ) is feasible, b ∗ i ( a ) ∈ [ b i ( a ) , b max ] for all a ∈ A and therefore b ∗ i ( · ) is also feasible. Furthermore, ˆ b i ( a ∗ ) = ϕ i ( a ∗ ) = b ∗ i andtherefore b ∗ i ( a ∗ ) = b ∗ i . Finally, I observe that u ( a ∗ , b ∗ i ( a ∗ ) , b o − i ( a ∗ )) = u ( a ∗ , b ∗ i , b o − i ( a ∗ )) = u ( a, ϕ i ( a ) , b o − i ( a )) ≥ u ( a, b ∗ i ( a ) , b o − i ( a ))for all a ∈ A. The last inequality follows from the definition of b ∗ i ( · ) which im-plies that b ∗ i ( a ) ≥ ϕ i ( a ) . This chain confirms that a ∗ ∈ arg max a ∈ A u ( a, b ∗ i ( a ) , b o − i ( a ))which concludes the proof for necessity.37ufficiency:Assume that ( a o , b oi ( a o )) is a feasible pair that solves the maximization problemin condition (ii) of the lemma but is not part of an equilibrium and that b oi ( · ) is afeasible bidding function. Then, there exists a feasible pair ( a ∗ , b ∗ i ( · )) such that:a) a ∗ ∈ arg max a ∈ A u ( a, b ∗ i ( a ) , b o − i ( a )) andb) u i ( a ∗ , b ∗ i ( a ∗ ) , b o − i ( a ∗ )) > u i ( a o , b o ( a o ))However, in this case u ( a ∗ , b ∗ i ( a ∗ ) , b o − i ( a ∗ )) ≥ u ( a, b ∗ i ( a ) , b o − i ( a )) ≥ u ( a, b max , b o − i ( a )) . The first inequality is due to (a) above, while the second inequality is because b ∗ i ( · )is feasible and therefore b ∗ i ( a ) ≤ b max for all a ∈ A. As a result, ( a ∗ , b ∗ i ( a ∗ )) satisfiesthe constraints in condition (ii) of the lemma, which implies that ( a o , b oi ( a o )) doesnot solve the maximization problem. This contradiction concludes the proof fornecessity and the lemma.Now I proceed with the rest of the proof.Let ( a o , b o ( · )) be an equilibrium of the game. Consider principal i. I will show thatthe inequality in condition (ii) of the lemma, holds as an equality. Assume the con-trary. Then, if the equilibrium bid equals the maximum bid or b oi ( a o ) = b max , thecontradiction is obvious. If on the other hand b oi ( a o ) < b max , the pair ( a o , b oi ( a o ))does not solve the maximization problem in the lemma. Indeed, in this case, thereexists b ∗ i such that b max > b ∗ i > b oi ( a o ) . Then, the feasible pair ( a o , b ∗ i ) satisfies theinequality constraint in condition (ii) of the lemma because b max > b ∗ i and yields u i ( a o , b ∗ i , b o − i ( a o )) > u i ( a o , b o ( a o )) because b ∗ i > b oi ( a o ) . In order to conclude theproof, I need to show that there exists a feasible biding function b ∗ i ( · ) such that b ∗ i ( a o ) = b ∗ i . In this respect consider the function b ∗ i = b oi ( a ) if a (cid:54) = a o b ∗ i if a = a o . Proof of 3B
The last argument proves proposition 3A. I turn now to propo-sition 3B.I start with a useful lemma.
Lemma: Maxima inequality.
Consider the feasible bidding functions b o ( · ) . Let a ´ ∈ arg max a ∈ A u ( a, b max , b o ( a ))38nd a o ∈ arg max a ∈ A u ( a, b o ( a )) . Then, u ( a o , b o ( a o )) ≥ u ( a ´ , b max , b o − i ( a ´)) . Proof:
Assume the contrary and consider any a ´ ∈ arg max a ∈ A u ( a, b max , b o ( a )) . Then, u ( a ´ , b max , b o ( a ´)) > u ( a o , b o ( a o )) ≥ u ( a ´ , b oi ( a ´) , b o − i ( a ´)) . Therefore, becausethe utility of the agent is strictly decreasing in bids it must be that b oi ( a ´) > b max , which is not possible because b o ( · ) is feasible. Thus, u ( a o , b o ( a o )) ≥ max a ∈ A u ( a, b max , b o ( a )) . I continue now with the rest of the proof. Assume that there exists a pair( a o , b o ( · )) as in the statement of proposition 3B, which is not a truthful equi-librium. Then, because ( a o , b o ( · )) is not an equilibrium, there exists an i and afeasible pair ( a ∗ , b ∗ i ( · )) such that b ∗ i ( a ∗ ) = b ∗ i , u i ( a ∗ , b ∗ i , b o − i ( a ∗ )) > u oi and a ∗ ∈ arg max a ∈ A u ( a, b ∗ i ( a ) , b o − i ( a )) . Then, the maxima inequality lemma above impliesthat u ( a ∗ , b ∗ i , b o − i ( a ∗ )) ≥ max a ∈ A u ( a, b max , b o − i ( a )) . Define u ∗ i = u i ( a ∗ , b ∗ i , b o − i ( a ∗ ))and b Ti ( · ) , which is the truthful response of i to b o − i ( · ) relative to u ∗ i . Then, b Ti ( a ∗ ) = b ∗ i . Therefore, u ( a ∗ , b Ti ( a ∗ ) , b o − i ( a ∗ )) ≥ max a ∈ A u ( a, b max , b o − i ( a )) = u ( a o , b o ( a o )) ≥ u ( a ∗ , b o ( a ∗ )) . For future reference I name this chain “principal chain”. The firstinequality in this chain is due to the maxima inequality lemma, the equality isbecause ( a o , b o ( · )) satisfies condition (Aii) and the last inequality because it sat-isfies condition (Ai). The principal chain implies that u ( a ∗ , b Ti ( a ∗ ) , b o − i ( a ∗ )) ≥ u ( a ∗ , b o ( a ∗ )) and because the utility of the agent is decreasing in all bids it followsthat b ∗ i = b Ti ( a ∗ ) ≤ b oi ( a ∗ ) . The last inequality along with the fact that the utilityof the principals is increasing in own bids implies that u i ( a ∗ , b oi ( a ∗ ) , b o − i ( a ∗ )) ≥ u i ( a ∗ , b Ti ( a ∗ ) , b o − i ( a ∗ )) = u ∗ i > u oi . Thus, u i ( a ∗ , b oi ( a ∗ ) , b o − i ( a ∗ )) > u oi , which yieldsfollowing the definition of truthful responses b oi ( a ∗ ) = b i ( a ∗ ) . Then, because b Ti ( · )is feasible it follows that b Ti ( a ∗ ) ≥ b i ( a ∗ ) = b oi ( a ∗ ) . Taken together b Ti ( a ∗ ) ≥ b oi ( a ∗ )and b Ti ( a ∗ ) ≤ b oi ( a ∗ ) imply that b Ti ( a ∗ ) = b oi ( a ∗ ) . As a result, u ( a ∗ , b Ti ( a ∗ ) , b o − i ( a ∗ )) = u ( a ∗ , b o ( a ∗ )) and therefore all inequalities in the principal chain hold as equalities.Thus, u ( a ∗ , b Ti ( a ∗ ) , b o − i ( a ∗ )) = u ( a ∗ , b oi ( a ∗ ) , b o − i ( a ∗ )) = max a ∈ A u ( a, b max , b o − i ( a )) = u ( a o , b o ( a o )) ≥ u ( a, b o ( a )) for all a ∈ A. In this chain the last inequality holds be-cause of condition (Ai).It follows that both a ∗ and a o maximize u ( a, b o ( a )) and sat-isfy conditions (Ai) and (Aii). Therefore, u oi = u i ( a o , b o ( a o )) ≥ u i ( a ∗ , b oi ( a ∗ ) , b o − i ( a ∗ )) =39 i ( a ∗ , b Ti ( a ∗ ) , b o − i ( a ∗ )) = u i ( a ∗ , b ∗ i , b o − i ( a ∗ )) = u ∗ i , which is a contradiction because u ∗ i > u oi . In the last chain the first inequality is because of condition (Bii) and theequalities because b Ti ( a ∗ ) = b oi ( a ∗ ) . Q.E.D.
B.1.4 Proof of proposition 6
The proof of plausibility and efficiency is the same as in section 2, up to thenecessary transformations of bidding and utility functions . However, the proofof calculation requires adjustment. Specifically, the related proof in section 2follows closely Dixit et al. (1997) and uses a lemma whose proof requires biddingfunctions to depend on the entire set A. For this reason, I provide here a newproof for private games. I only prove 3A, since the proof for 3B is very similar tothe proof in section 2.I consider again the case of market monotonicity. Assume that the pair( a o , b o ( · )) is an equilibrium and that u o is the vector of equilibrium utilities ofthe principals. First, I observe that from the maxima inequality lemma above(appropriately adapted) it follows that u ( a o , b o ( a o )) ≥ max a ∈ A u ( a, b max , b o − i ( a − i )) . I will proceed now to show that this expression holds as an equality. Assume thecontrary: u ( a o , b o ( a o )) > max a ∈ A u ( a, b max , b o − i ( a − i )) . Then, because the utility func-tions of the principals and agent are continuous with respect to bids, there exists b ∗ i ∈ R such that b max > b ∗ i > b oi ( a oi ) , u ( a o , b ∗ i , b o − i ( a o − i )) > max a ∈ A u ( a, b max , b o − i ( a − i ))and u i ( a oi , b ∗ i ) > u i ( a oi , b oi ( a oi )) . However, in this case, the pair ( a o , b o ( · )) is notan equilibrium. To see this consider the following deviation by principal i : b i = b max if a i (cid:54) = a oi b ∗ i if a i = a oi . Q.E.D. Except from the adjustment of utility and bidding functions, in the proof ofproposition 2 the expression “if a o ∈ A ´ ” must be substituted by “if thereexists an a ∈ A ´ such that a i = a oi ”.40 .2 Appendix to discussion B.2.1 Example 1
First I show that the unique symmetric efficient allocation is a = 1 , b = b =1 . Obviously in any efficient allocation a = 1 . Assume there exists an efficientallocation such that b i < i . Then, define µ = 1 − max { b , b } > . Consequently, if both the principals increase their bids at a = 1 by µ the utility ofboth the principals and the agent increases. Thus, in all efficient allocations a = 1and at least for one principal b i = 1 . In turn, this point implies that there is onlyone symmetric efficient allocation in which a = 1 and for both principals b i = 1 . Now I show that there are no efficient equilibria. Consider an efficient alloca-tion. In any such allocation a = 1 and at least for one principal b i = 1 . With-out loss of generality assume that b = 1 . Then there are two cases: b (1) > b (1) = 0 . If b (1) = λ > b = − λ if a = 10 if a (cid:54) = 1 . In this case the agent still chooses a = 1 and theutility of principal 1 increases regardless of his initial off equilibrium strategy.If instead b (1) = 0 principal 1 earns u = 0 . Then, principal 1 can deviate to b = 0 for all a ∈ [0 , . In this case, if there exists an a ∈ [0 ,
1] such that b ( a ) > u = a + 2 b > . If on the other hand b = 0 for all a ∈ [0 ,
1] the agent is indifferent among all a. Then, any a > a = 1 , principal 1 earns utility u = 1 > . Finally, as an example of an asymmetric inefficient equilibrium consider thefollowing: b = . a = 10 if a (cid:54) = 1 , b = . a = 0 .
20 if a (cid:54) = 0 . . These bidding func-tions yield a = 1 , b = 0 . , b = 0 , u = 0 . , u = 0 . , u = 1 . . B.2.2 Symmetric negative externalities
Examples 1 and 2 highlight the role of symmetry.
Example 1 u = b + b , u i = a − b i − b j ,a ∈ [0 ,
1] and b i (0) ∈ [0 ,
5] for both i . This game is symmetric and quasi-concave.Define allocation A as a = 0 and b = b = 1. For this allocation u = 2 and u = u = − . Graph 1 describes this situation. In this graph I , I and I are the indifferencecurves for the agent and the two principals that go through point A. The utility ofthe principals increases at allocations that lie to the south west of point A, whilethe utility of the agent at allocations that lie to the north east of the same point.In this regard the shaded area f depicts the allocations that make both principal 1and the agent better off than in A. In a similar manner, the shaded area g depictsthe same allocations for the agent and principal 2 . The fact that the areas f and g intersect only at point A implies conflict of interests. b b A I I I fg Graph 1
Symmetry and quasi-concavity.
Example 2
Let me now consider an example in which there is no conflict of interests. Considera variation of the previous game in which u = b + b , u = a − b − b and u = a − b − b . This game is quasi-concave but not symmetric. Define allocation42 as a = 0 , b = 1 and b = 1 . This allocation yields u = 2 , u = − u = − . Graph 2 reproduces graph 1 for this example. However, in this case, because ofasymmetry, there are allocations that make both principals and the agent betteroff than in A. The shaded area in graph 2 depicts these allocations. The existenceof such allocations violates conflict of interests. b b A I I I Graph 2
Quasi-concavity without symmetry.Now I turn to quasi-concavity. Examples 3 and 4 that follow highlight its role.
Example 3
Consider a game in which u = b + b , u = a − b − b , u = a − b − b , a ∈ [0 , b i (0) ∈ [0 ,
3] for both i . This game is both symmetric and quasi-concave.Define allocation A as a = 0 and b = b = 1 , which yields u = 2 , u = − u = − . Then, graph 3 depicts the respective indifference curves. As in graph 1the shaded areas f and g depict the allocations that make the agent and one ofthe principals better off than in A. Again, the fact that f and g intersect only in A, implies conflict of interests. 43 b A I I I g f Graph 3
Symmetry and quasi-concavity (2).I turn now to example 4.
Example 4
Consider a variation of example 3 in which u = b + b , u = a − b − √ b and u = a − b − √ b . This game is symmetric but not quasi concave. Graph 4 re-produces graph 3 for this example. The shaded area depicts the set of allocationsthat make both the principals and the agent better off than in A. The fact thatthis set is not empty violates conflict of interests.44 b A I I I Graph 4
Symmetry without quasi-concavity.
B.3 Appendix to applications
B.3.1 Market application
In order to solve this problem I use proposition 3B. First I observe that the lowerbound of profits is zero. This bound is always reached when the price also takesits lower bound which is equal to the average cost. Thus, this game satisfies strongdeep pockets and consequently condition (Bii). Therefore, an allocation of profits(Π i ) and quantities ( q i ) which satisfies conditions (Ai) and (Aii) gives rise to atruthful equilibrium.I proceed by solving the profit function with respect to p i which yields: p i = Π i + cq i q i ≡ ϕ ( q i ; Π i ) . I guess that this function is the equilibrium price functionaround the equilibrium quantities. In such a case, when both sellers offer thisprice function the agent faces the following problem:max q ∈ [0 , q ] − p q − p ( q − q )The unique solution to this problem is q i = q , which is independent of Π i . Thusfor this quantity condition (Ai) is satisfied.45 turn now to condition (Aii) which I use to calculate the equilibrium Π i . First,I notice that because of symmetry Π i = Π j ≡ Π . Then, I guess that when seller j submits the reservation price the buyer buys all the quantity from principal i .Under this guess, condition (Aii) becomes − − c q = − Π − cq , which in turnyields Π = Π i = c q . I substitute Π i in ϕ i ( · ) and get ϕ ( q i ) = cq q i + cq i . Then, inorder to verify my first guess I need to show that c q ≤ ϕ ( q ) ≤ p and in order toverify the second that ϕ ( q ) ≤ p. I proceed, by solving the equation p = c q + c ( q ∗ i ) q ∗ i with respect to q ∗ i in order tocalculate the quantity q ∗ i at which the price hits the upper bound. This exerciseyields: q ∗ i = p − √ p − c q c . In this way I obtain the price function p i ( · ) that Iprovide in the main text. Moreover, I find that ϕ ( q i ) < p for all q i > q ∗ i . Then,the assumption p > cq implies that q > q ∗ i and ϕ i ( q i ) ≥ cq i for all q i ∈ [0 , q ] ..