American Economic Journal Microeconomics
美国经济微观经济学杂志
A journal of the American Economic Association
美国经济学会的杂志
May 2012 VOLUME 4, NUMBER 2 Articles
To Review or Not to Review? Limited Strategic Thinking at the Movie Box Office
Alexander L. Brown, Colin F. Camerer, and Dan Lovallo
Ideologues Beat Idealists
Sambuddha Ghosh and Vinayak Tripathi
正文在此目录下
目录标题： Contracting with Heterogeneous Externalities 50到76页
作者： Shai Bernstein and Eyal Winter
Ignorance Is Bliss: An Experimental Study of the Use of Ambiguity and Vagueness in the Coordination Games with Asymmetric Payoffs
Marina Agranov and Andrew Schotter
Contracting in Vague Environments
Marie-louise Vier?
On the Robustness of Anchoring Effects in WTP and WTA Experiments
Drew Fudenberg, David K. Levine, and Zacharias Maniadis
Contractual and Organizational Structure with Reciprocal Agents
Florian Englmaier and Stephen Leider
Incentive Schemes, Sorting. And Behavioral Biases of Employees: Experimental Evidence Ian Larkin and Stephen Leider
正文第59页到67页
C. Optimal Ranking of Cyclic Tournaments
In the previous section, we demonstrated that optimal full implantation contracts are derived from a virtual tournament among the agents in which agentｉ beats agent ｊif ??(j)<??(i)).However, the discussion was based on the tournament being acyclic. If the tournament is cyclic, the choice of the optimal DAC contracting scheme (i.e., the optimal ranking) is more delicate since Lemma 1 does not hold. Any ranking is prone to inconsistencies in the sense that there must be a pair ｉ,ｊ such that ｉ is ranked above ｊ although ｊ beats ｉ in the tournament. To illustrate this point, consider a three-agent example where agent ｉ beats ｊ,agent ｊbeats ｋ, and agentｋbeats ｉ.The tournament is cyclic and any ranking of these agents necessarily yields inconsistencies. For example, take the ranking｛ｉ,ｊ，ｋ｝,which yields an inconsistency involving the pair(ｋ，ｉ)since ｋbeats ｉandｉis ranked above agent ｋ.This applies to all possible rankings of the three agents.
The inconsistent ranking problem is similar to problems in sports tournaments, which involve bilateral matches that may turn out to yield cyclic outcomes. Various sports organizations (such as the National Collegiate Athletic Association-NCAA) nevertheless provide rankings of teams/players based on the cyclic tournament. Extensive literature in operations research suggests solution procedures for determining the “minimum violation ranking” (e.g., Kendall 1962); Cook and Kress 1990; and Coleman 2005) that selects the ranking for which the number of inconsistencies is minimized. It can be shown that this ranking is obtained as follows. Take the
cyclic (directed) graph obtained by the tournament and find the smallest set of arcs such that reversing the direction of these arcs results in an acyclic graph. The desired ranking is taken to be the consistent ranking (per Lemma 1) with respect to the resulting acyclic graph.
One may argue that this procedure can be improved by assigning weights to arcs in the tournament depending on the score by which team ｉbeats teamｊand then look for the acyclic graph that minimizes the total weighted inconsistencies. In fact this approach goes back to Condorcet’s (1785) classical voting paper in which he proposed a method for ranking multiple candidates. In the voting game, the set of nodes is the group of candidates, the arcs’ directions are the results of pairwise voting, and the weights are the plurality in the voting. The solution to our problem follows the same path. In our framework arcs are not homogeneous and so they be assigned weights determined by the difference in the bilateral externalities. As in Condorcet’s (1785) voting paper, we will look for the set of arcs such that their reversal turns the graph into an acyclic one. While Young (1988) characterized Condorcet’s (1785) method axiomatically, our solution results form a completely different approach, i.e., the design of optimal incentives to maximize revenues.
Formally, we define the weight of each arc(ｉ,ｊ) ∈ A by t(i,ｊ) =?? ? -??(?).Note that weights are always nonnegative as an arc(ｉ,ｊ) refers to a situation in which ｊfavors ｉmore than ｉfavorsｊ. Hence t(ｉ,ｊ) refers to the extent one-sidedness of the externalities between the pairs of agents. If an inconsistency in the ranking arises due to an arc(ｉ,ｊ), then this implies that agent ｊprecedes agent ｉdespite the fact that ｉbeats ｊ. Relative to consistent rankings, inconsistencies generate additional costs for the principal. More precisely, the principal has to pay an additional t(ｉ,ｊ) when inconsistency is due to arc(ｉ,ｊ) ∈A.
For each subset of arcs S = {(?1,?1), (?2,?2),…，(??,??)} we define t(S)= (?,?)∈??(?,?),
Which is the total weight of the arcs in S .For each graph G and subset of arcs S we denote by G＿s the graph obtained from G by reversing the arcs in the subset S. Consider a cyclic graph G and let S* be a subset of arcs that satisfies the following:
(i) G-s* is acyclic.
(ii) t(S*)≤t(S) for all S such that G-s* is acyclic.
Then, G-s* is the acyclic graph obtained from G by reversing the set of arcs with the minimal total weight, and S* is the set of pairs of agents that satisfies inconsistencies in the tournament ranking of G –s*. Proposition 4 shows that the optimal ranking of G is the tournament ranking of G –s* since the additional cost from inconsistencies, t(S*), is the lowest.
PROPOSITION 4: Let (N,w,c) be a participation problem with a cyclic tournament G. Let φ be the tournament ranking of G-s*. Then, the optimal full implementation contracts are the DAC with respect toφ.
In the symmetric case, the principal cannot exploit the externalities among the agents, as ?????=0, and the total payment made by the principal is identical for all rankings. This can be seen to follow from Proposition 4 as well by noting that the tournament has two-way arcs connecting all pairs of agents, and t(i,j)=0 for all i ,j and t(S) is uniformly zero. An intriguing feature of the symmetric case is that all optimal contracting schemes are discriminative in spite of the fact that all agents are identical.
COROLLARY 4: When the externalities structure w is symmetric then all DAC contracts are optimal.
We can now provide the analogue version of Proposition 3 for the cyclic case. In this case, the
optimal ranking has an additional term ???????=t(S*) representing the cost of making the tournament acyclic, i.e., the cost borne by the principal due to inconsistencies.
PROPSITION 5: let (N, w, c) be a participation problem. Let ?????be the principal’s optimal cost of a full implementation contract. Then ?????=n · c - 1 2(????+?????) + ???????.
Corollary 3.1 still holds for pairs of agents that are we not in S*. More specifically, if we increase the level of asymmetry between pairs of agents that are outside S*, we reduce the total expenses that the principal incurs in the optimal contracting scheme.
Ⅲ.Extensions
In this section, We discuss the implications of the assumptions we made so far .We demonstrate that the optimal contracts remain optimal if we assume sequential participation choices when the principal desires to implement participation in a subgame perfect equilibrium with the property that each player has a dominant strategy on the subgame that he plays. In addition, We show that even when the outside option is affected by the agents' participation choices, the construction of the optimal contracts remains unchanged. We demonstrate that when contracts can be contingent on the participation of a subset of the agents, then the optimal contracts are closely related to the analysis above. Our analysis is valid in more general setups in which externalities can be either negative or positive. Moreover, the solution is also relevant to nonadditive externalities structures.
A. Sequential Participation Decisions
We first point out that our analysis applies to any sequential game except for one of perfect information, i.e., when each player is fully informed about all the participation decisions of his predecessors. Indeed, this extreme case of perfect information is a strong assumption as agents rarely possess the participation decisions of all their predecessors. Any partial information environment implies that some actions are taken simultaneously, and therefore the divide-and-conquer contracting scheme and the virtual tournament apply.
Nevertheless, it is interesting to point out that our analysis is also relevant to the extreme case of perfect information. Consider a game in which players have to decide sequentially about their participation based on a given order. Suppose that the principal wishes to implement the full participation in a subgame perfect equilibrium with the additional requirement that each player has a dominant strategy on the subgame in which he has to play. It is easily verified that the optimal contracting scheme in this framework is the DAC applied to the order of moves; i.e., the first moving player is paid c and the last player is paid c- ?∈???(j).Under this contracting scheme each player has a dominant strategy on each subgame. Assume now that the principal can control the order of moves (which he can do by making the offers sequentially and setting a deadline on agents' decisions).Then the optimal sequential contracting scheme is exactly identical to the one discussed in previous sections for the simultaneous case. If the principal suffices with a standard subgame perfect equilibrium(without the strategy dominance condition ) , then the optimal contracting scheme will allow him to extract more and he will pay c- ?∈???(j) to all agents.
B. Participation-Dependent Outside Options
In many situations, nonparticipating agents are affected by the participation choices of other agents. Consider the case of a corporate raider who needs to acquire the shares of N identical shareholders to gain control(similar to Grossman and Hart 1980).If the raider is enhancing the value of the firm when he holds a larger stake in the firm, then selling shareholders impose
positive externalities on nonparticipating agents. If the raider gains private benefits from the firm which will decrease its value, then selling shareholders induce negative externalities on the nonparticipating agents.
In this section we consider the case in which the agents' outside option is partly determined by the agents who choose to participate. For a given group of agents P≤N who participate, we define the outside option of nonparticipants as c+ ?∈???(j).In the former analysis we assumed η=0.22 Segal (2003) defines externalities as increasing (decreasing) when an agent is more (less) eager to participate when more agents participate. In our setup, eagerness to participate is identity-dependent. When η≤1, we say that agents are more eager to participate when highly valued agents choose to participate. Ifη>1 ,the benefit of nonparticipation outweigh the benefits of participation when highly valued agents choose to participate; hence agents are less eager to participate. In Segal's terminology, the former case is equivalent to increasing externalities, while the latter is equivalent to decreasing externalities.
Following the analysis of Proposition 1, if v is an optimal full implementation contracting scheme then it is easy to verify that under the current setup, v is a DAC of the form:
V=(c, c-(1-η)??2(?1),…,c-(1-η) ????(?? )),
Where φ=(?1,?2,…,??) is an arbitrary ranking. Instead if η>1, the participation problem is identical to a standard participation problem(with fixed outside option) where externalities are (1-η)??(j)<0. In these negative externalities problems the DAC mechanism does not apply and the optimal scheme requires that the principal reimburse the agents for their total burden, i.e., c-(1-η) ???(j),which is a positive number whenever the outside option and ??(j) are positive. Finally, the case of η=1 corresponds to an environment of no externalities at all and the optimal scheme requires simply to reimburse agents for their outside option. We can summarize with the following proposition:
PROPOSITIN 6: LET ( N, w,??) be a participation problem where ???=c+ ?∈???(j) and P≤N is a group of participating agents. LET G (N,A)be the equivalent tournament. The optimal full implementation contracts are given as follows:
(i)
(ii) For η<1,DAC contracts with respect to the optimal ???????;23 For η≥1, the optimal mechanism pays agent I the payoff c-(1-η) ???(j),
Which is exactly c, wherever η=1.
Note that one could consider a different case in which the outside option of agents in a linear function of the externalities agents induce. Also in this case, asymmetry improves the principal’s rant extraction.
C. Contingent contracts
Our model assumes that the principle cannot write contracts that make a payoff to an agent contingent on the participation of other agents. Under such contracts the principal could extract the total surplus form positive externalities among the agents. We find such contracts not very descriptive. Based on the data used by Gould, Pashigian, and Prendergast (2005)which consists of contractual provisions of over 2500 stores in 35 large shopping malls in the United States, there is no evidence that contracts make use of such contingencies. Shopping malls are a natural environment for contingent; the fact that these contracts are still not used makes it likely that in other, more complicated settings such contracts are exceptional as well. The theoretical foundation
for the absence of such contracts is beyond the scope of this paper. However, one possible reason for their absence is the complexity of such contracts, especially in environment s where participation involves long-term engagement and may be carried out by different agents at different points in time. We point out that if partial contingencies are used, i.e., participation is contingent on a subset of the group , our model and its analysis remain valid. Specifically, for each player i, let ?? ? N be the contingency set, i.e., the set of agents whose participation choice can appear in the contract with agent i. Let T= (?1,?2,?,??) summarize the contingency sets in the contracts. The optimal contract (when contingencies are not allowed). More precisely, let w be the original matrix of externalities. Denote by ?? the matrix of externalities obtained from w by replacing ??(?)with zero whenever j∈??. lemma 6.1 in the appendix shows that the optimal full implementation contracting scheme is as follows: agent I gets c if one of the agents j ∈?? does not participate ,i.e., the contingency requirement is violated. If all agents in ?? participate, then agent I gets the payoff ??(?,??,?) - ?∈???? ? , where ??(?,??,?) is the payoff for agent I for the participation problem (N,??,? ) under no-contingencies.
D.mixed externalities structure
So far we have limited our discussion to environments in which agents’ participation positively affects the willingness of other agents to participate. However, in many situations this is not the case, such as in environments of congestion. Traffic, market entry, and competition among applicants all share the property that the larger the number of agents who participate, the lower the utility of each participant. The heterogeneous property in our framework seems quite descriptive in some of these examples. In the conetxt of competition it is clear that a more qualified candidate/firm induces a large negative externality. It is also reasonable to assume, at lease for some of these environments, that the principal desires a large number of participants in spite of the negative externalities that they induce on each other.
In proposition 7, we demonstrate that in order to sustain full participation as a unique Nash equilibrium under negative externalities the principal has to fully compensate all agents for the participation of the others.
PROPOSITION 7: let (N, w, c)be a participation problem with negative externalities. Then optimal full implementation contracts v are given by ??=?+ ?≠?|??(?)| and v is unique.
Naturally, real-world multi-agent contracting problems ay capture both positive and negative types of externalities. In social events, individuals may greatly benefit from some of the invited guests, while preferring to avoid others. In a mall, the entry of a new store may benefit some stores by attracting more customers, but impose negative externalities on its competitors.
Our analysis of the mixed externalities case is based on the following binary relation. We say that an agent I is nonaverse to agent j if ??(?)≥0, and we write it as I≥j. we will assume that ≥ is symmetric and transitive, I.e., i≥j ? j≥I and if i≥j and j≥k then i≥k. note that this assumption does not imply and constraint on the magnitude of the externalities, but just on their sign. While the symmetry and transitivity of the nonaverse relation seem rather intuitive assumptions, not all strategic environments satisfy them. These assumptions are particularly relevant to environments where the selected population is partitioned into social, ethnic, or political groups with animosity potentially occurring only between groups but not within groups. We analyze a specific example of this sort of environment in Section IV.
It turns out that the optimal solution of participation problems with symmetry and transitivity of the nonaverse relation is derived by a decomposition of the participation problem into two separate participation problems: one that involves only positive externalities, and the other that involves only negative externalities. This is done by simply decomposing the externalities matrix into a negative and a positive matrix. In the following proposition we show that the decomposition contracting scheme, a contract set that is the sum of the two optimal contracts of the two decomposed participation problems, is the optimal contracting scheme for the mixed externalities participation problem.
Proposition 8: consider a participation problem (N, w, c).let (N,w+, c) be a participation problem such that ??+ ? =??(?) if ??(?)>0 and ??+ ? =0 if ??(?)<0 ,and let ?+be the optimal full implementation contracts of (N,?+ ,c). Let (N,?? , 0) be a participation problem such that ??? ? =?? ? if ??(?)<0and ??? ? =0 if??(?)≥0, and let ?? be the optimal full implementation contracts of (N, ??,0). Then, the decomposition contracting scheme v=?++?? induces a unique full participation equilibrium. Moreover, if agents satisfy symmetry and transitivity with respect to the relation, v is the optimal contracting scheme.
Proposition 8 shows that the virtual popularity tournament discussed in earlier sections plays a central role also in the mixed externalities case as it determines payoffs for the positive component of the problem. When symmetry and transitivity hold, the principal can exploit the positive externalities to reduce payments. In this tournament, i beats j whenever(i)?? ? ≥0 and ??(?)≥0, and (ii)??(?)>??(?). Note that, under the nonaverse assumptions, the principal provides complete compensation for the agents who suffer from negative externalities, as in the negative externalities case. Finally, it is easy to show that equivalently to Proposition 5, the principal’s cost of achieving full implementation in a mixed externalities setting is equivalent to the positive externalities setup, except that now the principal has to add the compensation for the negative externalities.
E. Nonadditive Preferences
We propose here an extension of the model in which preferences are no longer assumed to be separably additive. Using an iterative procedure that makes use of the solution for the additive case allows us to narrow down the set of potential optimal incentive contracts, even when no structure is assumed.
A participation problem is described by a group of agent N and their outside option is equal to c, as noted previously. We assume a general externalities structure, which is composed of the nonadditive preferences of the agents over all subsets of agents in the group N. More specifically, for each i, ??:2?{?}→R. The function ?? ? stands for the benefit of agent I from the participation with the subset S ? N. We normalize v(φ)=0. The condition of positive externalities now reads: for each i and sunsets S,T such that T?S we have ?? ? ≥??(?).
Arguments similar to those used in Proposition 1 show that the optimal contracting scheme that sustains full participation as a unique equilibrium also satisfies the divide-and-conquer property. Hence, the optimal contracts rely on the optimal ranking of the agents.
We leave the detailed description of the procedure to the proof to the Proposition 9. Instead, we provide an example to illustrate the basic ideas.
A Simple Example. -Consider a four-agent example. Given that the optimal solution is DAC for any ranking of agents φ = {?1,?2,?3,?4}, the DAC contracts with respect to ranking φ are (c, c-??2 ?1 ,????3 ?1,?2 ,????4(?1,?2,?3 )). Instead of identifying the optimal ranking. We
apply an iterative procedure of N-1 steps to eliminate rankings that we infer cannot be optimal. Our starting point is the set of all possible ranking of the agents; in this example there are 24 such ranking.
1 → 2
↑ ↗↘↓
3 → 4
Step 1:Let’s assume that the bilateral externalities ??(?)between the agents result in the corresponding acyclic graph described below.Therefore the tournament yields the unique consistent ranking for step one when Φ1= (3,1,2,4).
We argue that any ranking that orders the first two agents in a way that contradicts their relative ranking in Φ1 cannot be the optimal ranking. To see this, consider the ranking (4,2,1,3),which is consistent with Φ1 with respect to the relative ranking of agents 4 to 2.We can immediately construct a cheaper ranking by reversing the position of the first two agents, and keeping the position of the remaining agents e=ranked lower in the same order. Hence, we can eliminate (4,2,1,3) from the set of potential optimal ranking. Applying this logic to the entire set of potential rankings we are left with 12 potential rankings; i.e., the optional ranking of the original problem must start with any of the following pairs:(3,1),(3,2),(3,4),(1,2),(1,4),(2,4).
Step 2:We now proceed to the second iteration in which for each agent located the first position we construct a graph that is based on the bilateral relations conditional on the participation of the first agent .In particular, we consider the case in which agent 1 is ranked first and build the graph based on agents’ preferences conditional on the participation of agent 1;i.e.,the externalities matrix is given by (?? ? =??(?,1)|j∈{2,3,4}).
Let’s assume that preferences take the following forms:
?2(3,1)>?3(2,1);
?2 4,1 >?4 2,1 ;
?3(4,1)>?4(3,1).
Since the graph is acyclic the unique consistent ranking of the second iteration, conditional on agent 1 being first, is φ2|1=(4,3,2). Again, we require rankings to be consistent with φ2|1.For example, ranking(1,2,4,3)cannot be optimal since (2,4,3) is not consistent with φ2|1 and transposing the order of 2 and 4 we get ranking(1,4,2,3),which is cheaper. While there are six ranking in which agent 1 is ranked first, we can immediately eliminate three that do not agree with φ2|1 and we are left with {(1,4,2,3),(1,4,3,2),(1,3,2,4)}.However, these rankings must agree with the constraints from the previous step. This is not the case for ranking (1, 3, 2, 4),as we can transpose the order of 1 to 3 and get a cheaper mechanism; thus we can eliminate it as well.26Hence,if the optimal ranking starts with agent 1 it must be followed by agent 4 ranked second. Rather than discussing the construction of cases where agents 2 and 3 are ranked first, we continue to explore the case where agent 1 is ranked first and proceed to step 3.
Step 3:In this iteration we repeat and construct the graph based on agents 2 and 3’s preferences, conditional on the participation of agents 1 and 4.Let’s assume that ?2(3,1,4)<?3 (2,1,4);hence
φ3|1,4={2,3}.Thus, the only ranking that can be optimal in the original problem conditional on agent 1 being first is (1, 4, 2, 3).
General Result.-The example above illustrates our procedure for generating the optimal incentive contracts can also be used iteratively to eliminate no optimal rankings, when we impose no structure on agents’ preferences.
The starting point is the set of all agents’ rankings. We proceed with an iterative procedure of N-1 steps; at each step rule out possible rankings by constructing a graph that is based on the bilateral preferences of agents conditional on the participation of agents ranked about them. We assume that in each step the resulting graph is acyclic and thus generates a unique consistent ranking. We eliminate rankings that are inconsistent with the step’s consistent ranking or with the constraints imposed in the previous step. The formal description of this iterative procedure is provided in the proof of Proposition 9.
PROPOSITION 9: Let (N, c) be a participation problem with nonadditive preferences, for which all tournaments in the iterative procedure are acyclic. Then, the set of surviving ranking is nonempty and includes the optimal ranking.
Proposition 9 demonstrates that the fundamental logic underlying our analysis of additive externalities also underlies our construction, while taking into account the complex structure of externalities among agents.
Ⅳ .Group Identity and Selection
In this section we consider special externalities structures to demonstrate how the selection stage can be incorporated once we have solved the participation problem. Assume that the externalities take values of 0 or 1.In this environment an agent either benefits from the participation of peer or gains no benefit. We provide three examples of group identities in which the society is partitioned into two groups and agents have hedonic preferences for members in these groups. We demonstrate how the optimal contracting scheme proposed in previous sections may affect the selection of the planning of the initiative.
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