Indescribable Contingencies versus Unawareness and Incomplete Contracting

Indescribable Contingencies versus Unawareness and Incomplete Contracting Wenjun Ma Burkhard C. Schipper Job Market Paper November 4, 204 Abstract Maskin and Tirole (999) postulated that even though agents may not foresee physical properties of future contingencies, they are at least able to precisely foresee payoff consequences. They proved that such an indescribability of physical contingencies does not constrain contracting under symmetric information in the well-known buyer-seller model. We introduce a buyer-seller model with asymmetric information throughout the contracting relationship. We generalize Maskin and Tirole s result to asymmetric information with an extension of Bayesian implementation that allows for uncertainty also about sets of available outcomes. Our result for indescribable contingencies and asymmetric information is then contrasted with asymmetric unawareness under which agents may not precisely foresee some payoff consequences. We show with a simple example that unawareness of contingencies does constrain contracting. Keywords: Incomplete contracts, unforeseen contingencies, unawareness, asymmetric information, Bayesian implementation, strong implementation, augmented revelation mechanism, mechanism design, buyer-seller model. JEL-Classifications: C72, D78, D82. Burkhard is grateful for financial support from the National Science Foundation grant SES-06478. We thank Takashi Kunimoto for helpful discussions of the literature and Giacomo Bonanno for corrections. Department of Economics, University of California, Davis, Email: wma@ucdavis.edu Department of Economics, University of California, Davis, Email: bcschipper@ucdavis.edu

Introduction Real economic activity is governed by a fine grid of formal and informal contracts between agents in the economy. Most of these contracts are quite incomplete in the sense that they do not mention explicitly what each contracting party is supposed to do in all kinds of circumstances that may turn out relevant to the contracting relationship. The reasons for why contracts may be incomplete are summarized in Hart and Moore (988, p. 755): The difficult task facing the drafters of a contract is to anticipate and deal appropriately with the many contingencies which may arise during the course of their trading relationship. Since it may be prohibitively costly to specify, in a way that can be enforced, the parties are in practice likely to end up writing a highly incomplete contract. This casual observation lead to the development of the literature on incomplete contracts bringing together various examples on topics such as corporate governance, takeovers, property rights, the hold up problem etc. (For a survey, see Aghion and Holden, 20.) This literature has been criticized for failing to demonstrate that the inability to anticipate physical contingencies is indeed welfare relevant. In particular, Maskin and Tirole (999, p. 84) postulated that (i)f parties have trouble foreseeing the possible physical contingencies, they can write contracts that ex ante specify only the payoff contingencies. Then, later on, when the state of the world is realized, they can fill in the physical details. They further point out that if agents were unable to foresee at least payoff contingencies, then this would be at odds with the assumption that they are able to dynamically optimize (Maskin and Tirole, 999, p. 88), an assumption that is always imposed when applying subgame-perfect implementation in the literature on incomplete contracting. Maskin and Tirole (999) show in the context of a prototypical buyer-seller hold up problem that the indescribability of physical contingencies does not matter for welfare if agents satisfy the above mentioned Maskin-Tirole postulate. In other words, if agents follow the Maskin-Tirole postulate then they can sign contracts that may appear quite incomplete in the sense of not explicitly describing all relevant physical contingencies. Yet, these contracts are still complete in the sense of taking care of all payoff contingencies and achieve the same welfare as complete contracts under fully describable contingencies. Maskin-Tirole s result has been casually interpreted as saying that unforeseen contingencies do not matter for contracting. In this paper we will scrutinize to what extent this is really the case. Maskin and Tirole (999) show their result in a buyer-seller model with symmetric information at the stage of agreeing to the contract and complete information at the stage of 2

trading the good. These assumptions are standard in the literature. Yet, in reality most trading relationships take place under asymmetric information. Thus, one wonders whether or not indescribability of contingencies constrains welfare in contracting relationships in the presence of asymmetric information. Although Maskin and Tirole (999, Fn. 6) conjecture that the indescribability of contingencies does not matter for welfare even in the case of asymmetric information, it is far from clear how their model could be extended to show this formally. First, models of incomplete contracts under asymmetric information are rare in the literature; exceptions include for instance Bester and Krähmer (202) and Schmitz (2002). Second, to show their result Maskin and Tirole (999) translate physical contingencies into numbers and consider payoff functions for which sets of numbers are the domain. When asymmetric information is considered, agents may not only have different opinions over which sets of outcomes are available but also how they might be translated into numbers. Indeed, extensions of Maskin and Tirole s result to asymmetric information have proved to be challenging. First, Kunimoto (200) considers generalizing Maskin and Tirole s result to symmetric information in the contracting stage and asymmetric information in the trading stage. But as we show in Appendix A his assumptions imply that sets of outcomes are common knowledge among agents. In other words, he assumes away asymmetric information about the set of physical contingencies, which consequently allows him to apply standard results from Bayesian implementation by Mookerjee and Reichelstein (990). We believe that in order to convincingly argue that indescribability of physical outcomes does not matter for welfare in contracting, one needs to also allow for asymmetric information over available physical outcomes because state-dependent sets of physical outcomes are an important feature of the Maskin-Tirole setting and seem also relevant for contracting problems in reality. Second, Kunimoto (2008) considers a buyer-seller example with asymmetric information at the contracting stage and claims that under these assumptions Maskin-Tirole s result cannot be extended and indescribability becomes welfare relevant. Yet, as we show in Section 2. we can still devise a contract for his example that does not make reference to physical contingencies and achieves the same welfare as the date- interim efficient complete contract. Thus, it is not clear from the literature whether or not Maskin-Tirole s result can be extended to asymmetric information. Our contributions are as follows: First, we introduce a quite general buyer-seller model with asymmetric information throughout the contracting relationship. We show in a precise sense that for every such a buyer-seller model with describable physical contingencies there is an isomorphic buyer-seller model with indescribable contingencies. Second, we generalize Maskin and Tirole s result to asymmetric information throughout the contracting relationship. To prove this results, we extent Bayesian implementation à la Mookerjee and Reichelstein (990) from just incomplete information about payoffs to include also incomplete information about sets of available outcomes. Third, we take issue with Maskin-Tirole s postulate that agents are magically able to exactly forecast payoff contingencies even when they cannot forecast 3

physical contingencies (which among others also means that risk attitudes are not affected by the fact that physical contingencies are indescribable). We show by example that unawareness of contingencies is welfare-relevant despite the fact that agents are otherwise fully rational. Intuitively, an agent is unaware of a contingency if he does not even conceive of the contingency, the contingency is out of mind, or he does not pay attention to it and does not even realize that he does not pay attention to it. Taken together, our findings suggest the following interpretation: Unforeseen contingencies are welfare-irrelevant for contracting as long as agents are sufficiently aware of these contingencies so that they can precisely forecast payoff consequences. Yet, unforeseen contingencies become welfare-relevant for contracting when agents are unaware of some contingencies. The paper is organized as follows: In Section 2. we present Kunimoto (2008) counterexample to the generalization of Maskin and Tirole (999) to asymmetric information at the contracting stage. In Section 2.2 we show why it is not a counterexample. In Section 2.3 we present a simple example demonstrating that asymmetric unawareness of contingencies is welfare-relevant in contracting. In Section 3 we introduce our model of contracting under asymmetric information with describable contingencies. This is contrasted with the model under indescribable contingencies in Section 4. The main result is contained in Section 5. In Section 6 we discuss the interpretation of our result and the relationship to the literature. In Appendix B we present a generalization of Bayesian implementation to incomplete information about outcomes. All further proofs are relegated to Appendix C. 2 Illustrating Examples 2. The Counterexample by Kunimoto (2008) The purpose of restating the example by Kunimoto (2008) is threefold: First, we will adapt it to our notation and thus use it to illustrate our model. Instead of partitions, we use type-spaces to model asymmetric information throughout the contracting relationship. Moreover, instead of letting investments affect probability distributions over states of nature, we let investments affect payoffs. Second, we like to discuss why one may think that Maskin and Tirole s result cannot be extended to asymmetric information in the contracting stage. Third, we need to explain why Kunimoto s example is not a counterexample and illustrate why Maskin and Tirole s result can still be extended to asymmetric information. There are two agents. We denote the buyer by and a seller by 2. Both may agree that the buyer buys a good from the seller. The buyer may want to use this good as an input for his production. There are two different goods, ( triangle ) and ( circle ). First, we consider as a benchmark the case when contingencies and outcomes are describable throughout the contracting relationship. 4

The set of states of nature is T 0 = {t 0, t2 0, t3 0, t4 0, t5 0, t6 0, t7 0, t8 0, t9 0, t0 0, t 0, t2 0 }. The buyer has just one type. Thus his type space is a singleton, T = {t }. The seller s set of types is T 2 = {t 2, t2 2 }. The set of states of the world is T = T 0 T T 2. We assume that there is a common prior µ on T given in Table. Table : Common prior Common States of nature prior t 0 t 2 0 t 3 0 t 4 0 t 5 0 t 6 0 t 7 0 t 8 0 t 9 0 t 0 0 t 0 t 2 0 Agent 2 s t 2 2 2 2 2 2 2 0 0 0 0 0 0 types t 2 2 0 0 0 0 0 0 2 2 2 2 2 Agent s type t 2 At date 0, both agents learn their own type but not the type of the other or the state of nature. Each agent forms a belief about the state of nature and opponent s types. In our setting, agent s types determine their beliefs about states of nature and opponent s types via the common prior. Since agent has just one type, t, his date beliefs are given by the prior in Table. The date beliefs of agent 2 are shown in Table 2. Table 2: Date Beliefs of Agent 2 Agent 2 s States of nature beliefs t 0 t 2 0 t 3 0 t 4 0 t 5 0 t 6 0 t 7 0 t 8 0 t 9 0 t 0 0 t 0 t 2 0 Agent 2 s t 2 6 6 6 6 6 6 0 0 0 0 0 0 types t 2 2 0 0 0 0 0 0 6 6 6 6 6 6 Agent s type t At date, agents decide whether or not to accept a contract. The acceptance decision is observable by both agents. If at least one agent rejects the contract, then the game ends and every agent receives her outside option. The outside option gives a payoff of zero to both agents. At date 2, the buyer decides on a binary investment level, q {0, }, and the seller observes this decision. The cost of investment to the buyer is c(q) = if q =, and c(q) = 0 if q = 0. Irrespective of the investment level, the buyer learns the seller s type at the end of date 2. Thus, at date 3 the buyer s beliefs coincide with the seller s beliefs at date (see Table 2). Depending on the investment level, the seller receives additional information about states given by a partition and updates accordingly as shown in Table 3. At date 3, execute the trade prescribed the contract. The set of physical outcomes consists of the different goods to trade, and, as well as a no-trade outcome denoted by. Moreover, 5

Table 3: Date 3 Beliefs of Agent 2 Investment level States of nature q = t 0 t 2 0 t 3 0 t 4 0 t 5 0 t 6 0 t 7 0 t 8 0 t 9 0 t 0 Agent 2 s t 2 types t 2 2 2 2 4 4 4 4 2 2 4 Agent s type t Investment level States of nature q = 0 t 0 t 2 0 t 3 0 t 4 0 t 5 0 t 6 0 t 7 0 t 8 0 t 9 0 t 0 Agent 2 s t 2 types t 2 2 3 3 3 3 3 3 3 3 3 Agent s type t 0 t 0 t 2 0 4 4 4 0 t 0 t 2 0 3 3 3 it also specifies the price paid by the buyer to the seller which is denoted by y(t 0 ). This price may depend on the state of nature. We sometimes abuse of notation and denote the price simply by y. Besides types, we also introduce a set of payoff-types Θ = {θ, θ 2, θ 3 } that index agents ex post payoff functions. Both agent s ex post payoff functions, u i : Θ {,, } R R, i {, 2}, assign to each payoff state, physical outcome, and payment an ex post utility given in Table 4. The first entry in each cell represents the buyer s payoff while the second entry is the seller s. Payoff state θ 2 may be interpreted as the payoff state in which trading good is Table 4: Ex post payoff functions Payoff states θ θ 2 θ 3 u(,, y) ( y, y ) (5 y, y 0) ( y, y 0) u(,, y) ( y, y ) ( y, y 0) (5 y, y 0) u(,, y) ( y, y) ( y, y) ( y, y) efficient. Payoff state θ 3 is the payoff state in which trading good is efficient. In contrast, no trade is efficient in payoff state θ. States of nature and investment levels give rise to payoff-states according to a function p : T 0 {0, } Θ defined by p(t 0, ) = θ if t 0 {t 0, t2 0, t7 0, t8 0 } θ 2 if t 0 {t 3 0, t4 0, t5 0, t6 0 } θ 3 if t 0 {t 9 0, t0 0, t 0, t2 0 } 6

p(t 0, 0) = θ if t 0 {t 0, t2 0, t2 0, t7 0, t8 0, t9 0 } θ 2 if t 0 {t 4 0, t5 0, t6 0 } θ 3 if t 0 {t 0 0, t 0, t2 0 } The basic idea is that investment makes the desirable payoff types θ 2 and θ 3 more likely. When the states are describable at date, agents can sign a complete contract at date, which specifies for each state of the world in T the outcome as follows: 2 Date Contract Today we agree that at date 3 the seller will deliver at date 3 good if the state of nature is in {t 0, t 2 0, t 3 0, t 4 0, t 5 0, t 6 0}. Otherwise, she will deliver good. In return the buyer will pay at date 3 the amount of $8 to the seller. Buyer Seller We can represent this complete contract by the following social choice function f mapping states of the nature into outcomes: { (, 8) if t0 {t f(t 0 ) = 0, t2 0, t3 0, t4 0, t5 0, t6 0 } (, 8) if t 0 {t 7 0, t8 0, t9 0, t0 0, t 0, t2 0 } () If states of nature are verifiable (as sometimes assumed in the complete contract literature), then this contract can be implemented easily by let s say a mechanism in which the seller reports truthfully at date his type (i.e., the subset of states of nature to which he assigns strict positive probability) and that punishes him severely in court if he doesn t. If states of nature are not verifiable in court, then we need to consider whether agents have an incentive to reveal their information. Already Kunimoto (2008) observed that Contract satisfies the following four desiderata (see Appendix C for a detailed proof): Proposition Contract (i.e., the social choice function ()) satisfies the following four properties:. Date- incentive compatibility (henceforth denoted by IC ), In Kunimoto (2008) investments affect probabilities over states of nature. This is slightly different from our more explicit setting where states of nature and investments give rise to payoff types via the function p. This difference explains why in Kunimoto (2008) there are just 4 states of nature but 3 different payoff profiles while in our version of his example there are 3 payoff types but 2 states of nature. Essentially, states of nature are complete descriptions of physical events like how payoff profiles (3 payoff profiles) interact with investment levels (2 investment levels) and distributions of weights on payoff profiles (2 distributions) (i.e., 3 2 2 = 2). It is easy to verify that both in the model by Kunimoto (2008) and in our model, the weights attached to payoff profiles (θ, θ 2, θ 3) at date 3 are (, 2, 0) or (, 0, 2 ) if q = and (,, 0) or (, 0, ) if q = 0. All of our claims 3 3 3 3 2 2 2 2 about Kunimoto s example do not depend on whether we use his or our notation. 2 In line with the literature on incomplete contracting, we assume that investment levels are observable but not verifiable. Thus, the contract does not depend explicitly on the buyer s investment level. 7

2. Investment induced at date 2, 3. Date- participation constraints (henceforth denoted by P C ), and 4. Date- interim efficiency, i.e., the contract is efficient among all contracts satisfying IC. The complete contract f under describable contingencies can now be contrasted with the case of indescribable contingencies. Kunimoto (2008) claims that when contingencies and outcomes are indescribable, only no-trade contracts remain feasible and the buyer s optimal investment choice is not to invest. To see his argument, note first that as in Maskin and Tirole (999), contingencies and outcomes are indescribable at date but become describable at date 3. This motivates Kunimoto (2008) to consider only contracts satisfying date-3 incentive compatibility (henceforth denoted by IC 3 ) and date-3 participation constraints (henceforth denoted by P C 3 ). He observes the following (see Appendix C for a detailed proof): Proposition 2 The only contract satisfying date-3 participation constraints and date-3 incentive compatibility constraints is the no-trade contract with y = 0. Both agents receive an ex post payoff of 0, and the buyer makes no investment. This contract is date-3 interim inefficient among contracts satisfying date-3 incentive compatibility. Thus, if one follows Kunimoto s arguments that indescribability implies that both date-3 incentive compatibility and date-3 participation constraints are relevant, then indescribability matters under asymmetric information in the trading stage. 2.2 A Resolution to the Counterexample In this section, we show why Kunimoto s counterexample is not a counterexample. When contingencies and outcomes are indescribable, agents can still write and sign a contract at date that commits them to trade at date 3, without specifying which good to trade. In this case, we need to check whether P C and IC 3 can be satisfied. As long as P C is satisfied, agents would sign the contract at date even if it does not describe the contingencies and outcomes. Later at date 3, agents can use IC 3 to fill in the details of the contract. 3 An example of such a contract is the following. Date Contract 2 Today we agree that at date 3 the seller will deliver a good. Which good will be delivered will be at the discretion of the buyer. In return the buyer will pay the amount of $8 to the seller. Buyer Seller 3 We believe this is very much in spirit of Maskin and Tirole (999, p. 84) who write If parties have trouble foreseeing the possible physical contingencies, they can write contracts that ex ante specify only the possible payoff contingencies. (After all, it is only payoffs that ultimately matter.) Then, later on, when the state of the world is realized, they can fill in the physical details. The only serious complication is incentive-compatibility. 8

Clearly, Contract 2 does not mention any states or particular goods. Yet, it commits the seller already at date to deliver a good at date 3 to the buyer. Proposition 3 Contract 2 satisfies the following four properties:. Contract 2 implements the same outcome as Contract. I.e., Contract 2 implements the social choice function given in (). 2. Investment induced at date 2, 3. Date- participation constraints, 4. Date-3 incentive compatibility constraints. The proof is contained in Appendix C. Propositions and 3 together imply that Contract 2 is also date- interim efficient among contracts satisfying date- incentive compatibility. Agents are willing to participate in contract 2 at date, the buyer makes investment in equilibrium, and the outcome of Contract 2 is to trade if t 0 {t 0, t2 0, t3 0, t4 0, t5 0, t6 0 }, and if t 0 {t 7 0, t8 0, t9 0, t0 0, t 0, t2 0 } at the price $8. We note as an aside that in this example Contract 2 not only implement the same payoffs as Contract but also the same outcomes. This is stronger than what Maskin and Tirole (999) consider necessary for indescribability not to matter. We conclude that in this example indescribability does not matter, and Kunimoto s counterexample is not a counterexample to Maskin and Tirole s conjecture that indescribability continues to be irrelevant to incomplete contracting under asymmetric information. It turns out that this holds more generally. In Section 5 we prove the generalization of Maskin and Tirole (999) to asymmetric information throughout the contracting relationship. 2.3 A Simple Example of Unawareness in Incomplete Contracting The previous example should not let us believe that unforeseen contingencies are irrelevant for contracting. In the previous example, agents where still able to anticipate correctly outcomes and payoffs despite the fact that they were unable to describe outcomes in a contract. The purpose of the following example is to demonstrate briefly that, if instead of indescribable contingencies we model unforeseen contingencies by unawareness of contingencies, then unforeseen contingencies can become welfare-relevant in contracting. Under unawareness agents may not be able to foresee all payoff-relevant contingencies, possible kinds of investment choices, outcomes, or even agents. Here we focus on a simple example involving double-sided asymmetric unawareness of payoff-relevant contingencies. There are two agents, a buyer and a seller 2. For simplicity there is just one good and the no-trade outcome. Moreover, we abstract from any investment choices. The seller is aware of a novel feature of his good of which the buyer is unaware of. The buyer is aware of another novel application for this good that the seller is unaware of. It so happens that if the good has the novel feature then it becomes extremely valuable when applied in the novel way the buyer is aware of. Unfortunately, because of their asymmetric unawareness, both agents don t realize this and do not trade although trade would be efficient if agents could pool their awareness. More formally, there are four disjoint singleton type-spaces, the top space T = {(t 0, t, t 2 )}, the left space T = {( t 0, t, t 2 )}, the right space `T = {(`t 0, `t, `t 2 )}, and the bottom space 9

T = {(t 0, t, t 3 )}. These spaces form a complete lattice in the sense T and `T are incomparable, but both are below T and above T. T is the space where the novel application is expressible but not the novel feature of the good. `T is the space where the novel feature of the good is expressible but not the novel application. T is the space where both the novel application and the novel feature is expressible. Finally, T is the space where neither the novel application nor the novel feature are expressible. Since we focus for simplicity only on perfect information (and unawareness), we consider only singleton state spaces. Beliefs are modeled by µ (t )( t 0, t 2 ) = µ ( t )( t 0, t 2 ) =, µ (`t )(t 0, t 2 ) = µ (t )(t 0, t 2 ) = µ 2 (t 2 )(`t 0, `t ) = µ 2 (`t 2 )(`t 0, `t ) =, µ 2 ( t 2 )(t 0, t ) = µ 2 (t 2 )(t 0, t ) = This information structure is a special case of an unawareness beliefs structure à la Heifetz, Meier, and Schipper (203a). The information structure and payoffs are depicted in Figure. The first component of the payoffs written below each state refers to the buyer s payoff while Figure : (0 y; y 6) T (t 0 ; t ; t 2 ) T ( t 0 ; t ; t 2 ) (µt 0 ; µt ; µt 2 ) T µ (5 y; y 6) (5 y; y 6) (t 0 ; t ; t 2 ) T (5 y; y 6) the second component refers to the seller s payoff. Both payoffs depend of course on the price y. The beliefs are indicated by the blue solid and dashed ovals and arrows for the buyer and seller, respectively. It is straightforward to see that only the no-trade contract is implementable. The buyer lives in space T and believes the seller lives in space T, while the seller lives in fact in space `T and believes that the buyer lives in space T. Neither realizes that if they could pool their awareness trade is welfare-improving and implementable for instance with Contract 2. Without being aware of what the other agent is aware of, agents wouldn t want to participate in such a contract. Such a situation is perhaps symptomatic of the failure of interdisciplinary research collaborations where each party may have knowledge that becomes only valuable when pooled with the other party s knowledge. But since they have no idea about the other party s knowledge, they fail to realize how valuable their own knowledge is. 0

3 Describable Contingencies Our setting is analogous to Maskin and Tirole (999) except that we allow for asymmetric information throughout the contracting relationship. To this extent, we use type-spaces. There are two agents, a buyer and a seller. They may agree that the buyer buys a good from the seller. The buyer may want to use this good as an input for his production. Trade under a particular contract may make both parties better off in the sense that both parties can receive positive surplus, rather than zero surplus under no trade. We will consider two scenarios. First, we consider the benchmark case when contingencies and outcomes are describable throughout the contracting relationship. This will be contrasted with the case in which contingencies and outcomes are indescribable. 3. Timing of the Contractual Relationship Date 0 (Information stage): There is a nonempty finite set of states of the world T = T 0 T T 2. Space T 0 is a set of states of nature. Spaces T and T 2 are the sets of types of agents and 2, respectively. The profile t = (t 0, t, t 2 ) T denotes a generic state of the world. As usual we abuse notation slightly and denote by T i = T 0 T 3 i and write t i = (t 0, t 3 i ) T i, for i {, 2}. At date 0, both agents learn their own type but not necessarily the type of the other or the state of nature. Each agent forms beliefs about states of nature and opponent s types. To this extent, µ (T ) is common prior with t i T i µ(t i, t i ) > 0 for all t i T i and i {, 2}. We let µ i : T i (T i ) be a function that associates to each type of agent i a belief over states of nature and opponent s types. This belief is derived from the common prior by µ i (t i )(t i ) = µ(t i, t i ) t i T i µ(t i, t i ). Nature also proposes a contract f F from some nonempty finite set F. The purpose of nature proposing the contract is to avoid modeling the potentially complicated bargaining game. We assume that the choice of contract is independent from the selection of types and states of nature. The contract is completely uninformative about types and states of nature. Both agent s learn which contract has been selected. Thus, they do not need to model beliefs about it. Date (Contracting stage): Agents decide simultaneously and independently whether or not to accept the contract selected by nature at date 0. This decision will depend on the contract proposed by nature and the agent s type. Agent i s strategy at this stage is a i : T i F {Y es, N o}. We require that each agent s acceptance strategy is adapted to his information, i.e., a i (t i ) = a i (t i ) for all t i, t i T i such that µ i (t i ) = µ i (t i ).4 If at least one agent rejects the contract, then the game ends and every agent receives her outside option. If both agents accept the contract, then this becomes commonly known. In this case, each agent i updates her belief 4 As it will become clear shortly, payoffs are affected by states of nature. Any two types of an agent with the same beliefs have also the same beliefs about payoffs.

by µ i (t i )(t i f) = µ i(t i )({t i } (T 0 (a 3 i (f)) (Y es)) µ i (t i )(T 0 (a 3 i (f)), (Y es)) where (a 3 i (f)) (Y es) T 3 i is the set of opponent s types that agree to contract f given the acceptance strategy a 3 i. That is, beliefs are consistent with acceptance strategies. Date 2 (Investment stage): Each agent decides simultaneously and independently on an investment level. This decision will depend on the contract proposed by nature, the agent s type, and the opponent s acceptance decision at date. Yet, since we only reach date 2 when both agents accept the contract, we do not explicitly need to make the acceptance or rejection of a contract an argument of the investment choice. That is, the investment strategy of agent i is a function e i : T i F Q i, where Q i is agent i s nonempty set of investment choices. Again, we require that each agent s investment strategy is adapted to her information. That is, for all f F, e i (t i, f) = e i (t i, f) for all t i, t i T i such that µ i (t i )( f) = µ i (t i )( f). In line with the previous literature, we assume that investment levels q = (q, q 2 ) Q = Q Q 2 are observable to both agents after date 2 but non-contractible. We want to allow for investment into information (e.g., market research ). 5 To this extent, we let Π i (q) be agent i s partition over T i when investment levels are q. If agents choose investment levels q Q and t i T i is the true profile, then agent i is assumed to observe the partition cell π i (q) Π i (q) with t i π i (q) T i. Given the contract f and investment strategies e = (e, e 2 ) (which will be known in equilibrium; see below) and agent 3 i s observed investment q 3 i, agent i updates her belief at the end of date 2 by µ i (t i )(t i f, e, q 3 i ) = µ i (t i )({t i } (T 0 (a 3 i (f)) (Y es) (e 3 i (f)) (q 3 i )) π i (e i (t i, f), q 3 i )) µ i (t i )(T 0 ((a 3 i (f)) (Y es) (e 3 i (f)), (q 3 i )) π i (e i (t i, f), q 3 i )) where (e 3 i (f)) (q 3 i ) T 3 i is the set of opponent s types that would invest the amount q 3 i observed by i given investment strategy e 3 i. That is, each agent s belief is consistent with investment strategies (as well as with any other information observed by her). Date 3 (Trading stage): In the trading stage the agents take actions to execute the contract that implements the outcome and payments after which the game ends. There may be both uncertainty about the agent s preferences and the outcomes available. For each agent i {, 2}, there is a nonempty finite set of payoff-types Θ i. We denote by Θ = Θ Θ 2 the set of profiles of payoff-types. Moreover, we let θ = (θ, θ 2 ) denote a generic profile of payoff-types. For each agent i {, 2}, investments and state of nature give rise to payoff-types via a function p i : T 0 Q Θ i. 6 5 This is to be analogous to the standard framework in which investment affects the probability distribution over states. 6 In the standard contract literature, investment affects the probability distribution over payoff types. In our more general setting, investments and states of nature affect payoff types. (One can think of payoff types as investment-dependent random variables on the space of nature.) Since agents are uncertain about states of nature, they might be uncertain about payoff types or how investments affects payoff types. 2

Uncertainty about outcomes is modeled with a nonempty finite set of outcome states Ξ. If ξ Ξ is the true outcome state, then X(ξ) denotes the nonempty finite set of physical outcomes available at ξ. Denote X = ξ Ξ X(ξ). We can interpret x X as an allocation of physical goods among agents and 2 at date 3. States of nature and investments give rise to outcome-states via a function o : T 0 Q Ξ. Besides the outcome-state dependent set of physical allocations, there is an outcome-state independent set of outcomes Y. For concreteness, we shall assume that the choice of y Y specifies the distribution of money among agents. That is, y = (y i, y 3 i ), where y i is agent i s allotment of money, i {, 2}. We let Y := {(y, y 2 ) R : y i y i y i, i {, 2}, and y + y 2 0}. That is, subsidies from third parties are excluded. For each agent i {, 2}, the ex post payoff function u i : (t 0,q) T 0 Q {p(t 0, q)} X(o(t 0, q)) Y i R maps payoff types and outcomes (both consistent with the state of nature and investment levels) into payoffs. This formulation takes care of the fact that not all payoff types or outcomes are consistent with or available in every state of nature. We assume that u i is strictly increasing and continuous in y i Y i. Figure 2 provides an overview over the timeline of our contract setting. Figure 2: Timeline of the Model Date 0 Information stage Date Contracting stage Date 2 Investment stage Date 3 Trading stage Both agents learn their own type Both agents decide whether or not to accept the contract; becomes known Both agents decide on an observable investment; further information revealed Both agents execute the contract 3.2 Complete contracts A complete contract is a social choice function, i.e., a mapping f : T X Y such for all states of the world t = (t 0, t, t 2 ) T, f(t) X X(o(t 0, q)) for all q Q, where f(t) X refers the physical outcome component of the outcome f(t). That is, at every state of the world, the contract can only assign a physical outcome available for every profile of investments. We think it is somewhat misleading to speak of social choice functions as contracts. Contracts in the real world correspond rather to mechanisms (i.e., rules of interaction ) implementing social choice functions. Nevertheless, we will stick here to the terminology in the literature. 3

We can now write the date- interim expected payoff of type t i of agent i {, 2} from both agents accepting the complete contract f F and their investment strategies e as Ui (e, f, t i ) := µ i (t i )(t 0, t 3 i )u i (p(t 0, e(t i, t 3 i, f)), f(t i, t i )) c i (e i (t i, f)), (t 0,t 3 i ) T i where c i (q i ) is agent i s cost of investing q i Q i. 7 3.3 Investment Game With this notation in place, we can revisit the investment game of date 2. Assume that at date 3, agents implement the contract f. Then the contract induces a Bayesian investment game at date 2 in which agents choose investment strategies e = (e, e 2 ) and agent i s expected payoff function at type t i is given by Ui 2 (e, f, t i ) = µ i (t i )(t 0, t 3 i f)u i (p(t 0, e(t i, t 3 i, f)), f(t i, t i )) c i (e i (t i, f)). (t 0,t 3 i ) T i A pair of investment strategy profiles and a complete contract (e, f) is feasible if, given the complete contract f, the unique equilibrium of the Bayesian effort game consists of each agent i selecting the investment strategy e i (, f) such that (i) e (, f) constitutes a Bayesian Nash equilibrium, i.e., for all t i T i and i {, 2}, U 2 i (e, f, t i ) U 2 i (q i, e 3 i, f, t i ) for all q i Q i, (ii) there is no other Bayesian Nash equilibrium. The definition of feasibility is analogous to the prior literature except that we require unique Bayesian equilibrium rather than unique Nash equilibrium because we consider asymmetric information throughout. Denote by e (, f) the unique Bayesian Nash equilibrium of the Bayesian effort game associated with the complete contract f. Moreover, we denote by q = (q, q 2 ) the profile of investments realized in the unique Bayesian Nash equilibrium. The date-3 interim payoff of agent i expected from a complete contract f when her type is t i, investment strategies are in unique equilibrium, and q3 i is the opponent s investment observed by agent i in equilibrium is given by Ui 3 (q3 i, f, t i ) := µ i (t i )(t i f, e, q3 i)u i (p(t 0, e i (t i, f), q3 i), f(t i, t i )) c i (e i (t i, f)). t i T i 7 This formulation implicitly assumes that all types of the opponent would accept contract f. We will later impose the assumptions that date- interim participation constraints hold for all types. 4

4 Indescribable Contingencies The basic ideas of indescribability in Maskin and Tirole (999) are as follows: In the language of our model, payoff types in Θ should be thought of descriptions of payoffs while outcomes states in Ξ are descriptions of sets of outcomes. Indescribability by agents affects how agents can make use of these descriptions. First, although agents cannot describe outcomes or sets of outcomes, they are assumed to be at least able to describe the cardinality of outcomes. They are not able to describe Ξ though. They are also not able to describe for the outcome state ξ Ξ the set of available outcomes X(ξ) (or equivalently ξ) but just the cardinality of outcomes n = X(ξ). Second, agents can describe future payoffs. There is a problem of letting payoffs be describable but outcomes indescribable because θ can be understood as describing how physical outcomes in X are mapped into utilities (at state θ). That is, θ must implicitly describe outcomes in the domain of the utility mapping. As remedy, Maskin and Tirole (999) allow agents to translate outcomes into numbers and consider number-based payoff functions that are assumed to preserve von Neumann-Morgenstern preferences. That is, the resulting new payoff types θ Θ under the indescribability constraint describe how numbers are mapped into utilities. Since these number-based utilities are supposed to correspond to utilities from outcomes, we need to consider how outcomes are translated into numbers. There may be even asymmetric information about such translations of outcomes into numbers. Let Λ be a nonempty finite set of states describing possible translation keys or code books for the translations of outcomes into numbers. Further let l : T 0 Q Λ be a mapping that associates with each state of nature and investment a translation of outcomes into numbers. 8 That is, how outcomes are translated into numbers (i.e., the language, code book ) may depend on the state of nature and the investments. The translation is given by a mapping d : {l(t 0, q)} X(o(t 0, q)) N (t 0,q) T 0 Q that maps for each translation state outcomes consistent with this translation state into numbers in N N. 9 To implement the indescribability constraints, we must let each θ Θ be describable while neither Ξ nor a particular ξ Ξ are describable. ξ is describable only to the extent of the cardinality of outcomes available at ξ. λ cannot be describable. These indescribability constraints come on top of observability and verifiability assumptions of incomplete contracting. In particular, Maskin and Tirole (999) assume that θ and λ are both non-verifiable at date 3 while both Ξ and the realized outcome state ξ become verifiable in court. In contrast to the standard incomplete contracting literature, there is not necessarily a stage in our setting with asymmetric information where everything is commonly observed. The following table summarizes the assumptions. 8 l for language. 9 Maskin and Tirole (999) call the inverse of it the deciphering key. 5

Feature Parameter Describable Describable Verifiable space at date at date 3 at date 3 Payoff states Outcome states Translation keys Θ Ξ Λ We can now proceed to state the model with indescribable contingencies analogously to the model with describable contingencies. Rather than repeating every stage, we like to point out what changes as compared to the model with describable contingencies. From the exposition above it is clear that states of nature in T 0 model uncertainty about () payoffs, (2) available physical outcomes, and (3) translations of outcomes into numbers because states of nature in T 0 give rise to payoff types, available outcomes, and translation states via the mappings p, o, and l, respectively. If agents are assumed to be constrained in their ability to describe available outcomes and translations of outcomes into numbers, their types cannot be based as in Section 3 on states of nature in T 0 but must be based on a different space of states of nature in which states serve as descriptions of the situation subject to the indescribability constraints. Let T 0 denote such a space of states of nature. Further, we use hat above any mathematical object to denote the object in the indescribable case. E.g., T = T 0 T T 2 is the type space based on states of nature in T 0, µ ( T ) is the common prior on T etc. For each agent i {, 2}, there is a nonempty finite set Θ i that denotes payoff-types in the indescribable model and a function p i : T 0 Q Θ i. The cardinality of physical outcomes is a finite natural number in N. If the cardinality of physical outcomes is n, we let N(n) = {,..., n}. We interpret N(n) as the set of numbers representing physical outcomes when n is the cardinality of the set of physical outcomes. Clearly, N(n) = n. We let N := {,..., X }. States of nature and investments give rise to such less precise outcome -states via a function ô : T 0 Q N. We assume agents have no problems in describing the outcome-state independent set of outcomes Y since they represent just prices or money transfers. For each agent i {, 2}, the number-based ex post payoff function û i : ( t 0,q) T 0 Q { p( t 0, q)} N(ô( t 0, q)) Y i R maps payoff types in the indescribable model and numbers (both consistent with the corresponding state of nature in the indescribable model) into payoffs. A number-based contract is a social choice function, i.e., a mapping f : T N Y such for all states of the world in the indescribable model t = ( t 0, t, t 2 ) T, f( t) N N(ô( t 0, q)) for all q Q, where f( t) N refers to the number component of f( t). Number-based contracts differ from complete contracts in the model with describable contingencies both in their domain and codomain. The domain T is based on the space of nature T 0. Further, in the codomain, the set of outcomes is replaced by a subset of natural numbers. 6

Definition (Isomorphic model with indescribable contingencies) A model with indescribable contingencies is isomorphic to the model under describable contingencies if there exists a tuple of bijective functions ρ = (ρ i ) i {0,,2} with ρ i : T i T i for i {0,, 2} such that () ρ 0 satisfies for all q Q and all t 0 T 0, (.) ô(ρ 0 (t 0 ), q) = X(o(t 0, q)), (.2) d(l(t 0, q)) is a bijection between X(o(t 0, q)) and N(ô(ρ 0 (t 0, q)), (.3) for i {, 2}, the number-based payoff function satisfies that there exist α i (ρ 0 (t 0 ), q) > 0 and β i (ρ 0 (t 0 ), q) R such that for all x X(o(t 0, q)) and y i Y i, u i (p(t 0, q), x, y i ) = α i (ρ 0 (t 0 ), q)û i ( p(ρ 0 (t 0 ), q), d(l(t 0, q), x), y i ) + β i (ρ 0 (t 0 ), q).(2) (2) For all t T, µ(ρ(t)) = µ(t). We define an isomorphism between the two models with the existence of a bijection. That is, for each state of the world in the model with describable contingencies there is exactly one state of the world in the model with indescribable contingencies and vice versa. Recall that the state of nature determines payoff types, outcome states, and translation states. Thus, the mapping ρ 0 has to specify how these objects are related in both models. (.) means that for each profile of investments, the outcome state assigned by the state of nature in the model with indescribable contingencies is the cardinality of outcomes in the corresponding state in the model with describable contingencies. (.2) requires that for each profile of investments and state of nature, the translation key is a bijection between the set of available outcomes in the describable model and the set of numbers representing outcomes in the indescribable model. (.3) assumes that risk preferences are unchanged by indescribability. That is, for every profile of investments and every state of nature in the model with indescribable contingencies, the preferences are identical to the ones in the corresponding state of nature in the model with describable contingencies. That is, numbers and payments in the indescribable case are ordered as the corresponding physical outcomes and payments in the describable case. Although von Neumann-Morgenstern utilities are defined up to affine transformations, we are able to adopt the following simplifying convention: Remark For i {, 2} and every state of nature t 0 and profile of investments, we can define ûi ( p(ρ 0 (t 0 ), q), d(l(t 0, q), x), y i ) := α i (ρ 0 (t 0 ), q)û i ( p(ρ 0 (t 0 ), q), d(l(t 0, q), x), y i ) + β i (ρ 0 (t 0 ), q). and simply write (in analogue to Maskin and Tirole, 999, p. 88), u i (p(t 0, q), x, y i ) = û i ( p(ρ 0 (t 0 ), q), d(l(t 0, q), x), y i ) instead of equation (2). This convention will be adopted throughout and we omit tilde from û i. Property (2) of Definition states that the common prior beliefs in the model with describable contingencies and in the model with indescribable contingencies are consistent for corresponding states of the world. For beliefs of types this property implies the following (see Appendix C for a short proof): 7

Remark 2 For all t i T i and t i T i, µ i (ρ i (t i ))( t i ) = µ i (t i ) ( (ρ i ) ( t i ) ). We understand Maskin and Tirole (999) as suggesting that for each model with describable contingencies there is an isomorphic model with indescribable contingencies. With our notation in place we can formally prove this in the following proposition. Proposition 4 For every model with describable contingencies, there is an isomorphic model with indescribable contingencies. The proof in Appendix C is constructive. It reveals that the set payoff types in the model with indescribable contingencies must be richer than the one in the isomorphic model with describable contingencies. This is not surprising. Since number-based payoff functions cannot depend on outcome states but available outcomes are encoded into numbers, this information must be instead captured by the payoff uncertainty in the indescribable case. In fact, that s how we think the model captures the Maskin-Tirole postulate according to which agents must be at least able to foresee precisely payoff consequences (of all kinds of indescribable contingencies). Definition 2 The number-based contract f corresponds to a complete contract f in ex post utilities if for all t T and q Q, û i ( p(ρ 0 (t 0 ), q), f(ρ(t))) = u i (p(t 0, q), f(t)) for i {, 2}. This definition does not require the number-based contract to yield the numbers corresponding to the outcomes of the corresponding complete contract. (I.e., does not necessarily yield the same physical outcome subject to indescribability constraints.) Rather, it just requires to yield the same ex post utilities. This is a natural requirement under the Maskin-Tirole postulate of being able to foresee at least precise payoff consequences. In the following, we show that if a number-based contract f corresponds to a complete contract f in ex post utilities, then throughout the contracting relationship the incentives in the model with describable contingencies are preserved in the isomorphic model with indescribable contingencies. First we show that if a number-based contract corresponds to a complete contract in ex post utilities, then date- interim utilities of corresponding types are equal (see Appendix C for a proof). Lemma If the number-based contract f corresponds to a complete contract f in ex post utilities, then for every i {, 2}, t i T i, and q Q, Û i (q, f, ρ i (t i )) = U i (q, f, t i ). In the following text, we will assume analogous to Maskin and Tirole (999, Assumption ): Assumption (Existence of a describable no-trade outcome) There exists a describable alternative x 0 X and transfers y 0 Y such that for all t = (t 0, t, t 2 ) T and q Q, (i) x 0 X(o(t 0, q)), 8

(ii) u i (p(t 0, q), x 0, y 0 i ) = u i(p(t 0, q ), x 0, y 0 i ) for all q Q and i {, 2}, (iii) y i < y 0 i < y i for i {, 2}. (x 0, y 0 ) is the outcome if the contract is not accepted in stage. The outcome (x 0, y 0 ) may be interpreted as the no-trade outcome or outside option. If the contract is not accepted in stage, then typically no investments are made. Hence, the evaluation of (x 0, y 0 ) does not depend on investments q Q. It may still depend on the state of nature though. The date- interim payoff of type t i of agent i from the outside option is for any q Q, U i (x 0, y 0, t i ) := t 0 T 0 µ i (t i )(t 0 ) T0 u i (p(t 0, q), x 0, y 0 i ), where µ i (t i ) T0 denotes the marginal of µ i (t i ) on T 0. Note that from Assumption (ii) it follows that U i (x0, y 0, t i ) is constant in q Q. It may still vary with t i though. In the model with indescribable contingencies, we define Û i (x0, y 0, t i ) analogously. Clearly, we have Û i (x0, y 0, ρ i (t i )) = U i (x0, y 0, t i ) for all t i T i. Definition 3 (Date- Interim Participation Constraints) The complete contract f satisfies date- interim participation constraints for both agents i {, 2} and all types t i T i, if U i (e, f, t i ) U i (x 0, y 0, t i ) for all t i T i. Note that with this formulation agents take already into account the optimal investment upon all types of both agents accepting the contract. This is because optimal investment strategies are known in equilibrium. Note further that if f satisfies date- interim participation constraints then for both agents i {, 2} and all types t i T i, a i (t i, f) = Y es. Analogously to Definition 3, we define when a number-based contract satisfies date- interim participation constraints. The following lemma states that investment decisions are unaffected by indescribability constraints (see Appendix C for a proof): Lemma 2 Suppose that the number-based contract f corresponds to a complete contract f in ex post utilities. If (ê, f) and (e, f) are feasible and ê (ρ(t), f) = e (t, f) for all t T, then the complete contract f satisfies date- participation constraints if and only if the number-based contract f satisfies date- participation constraints. Conversely, if both the complete contract f and the number-based contract f satisfy date- interim participation constraints, then (ê, f) is feasible if and only if (e, f) is feasible and ê (ρ(t), f) = e (t, f) for all t T. The immediate consequence is that incentives at the beginning of the trading period in the model with indescribable contingencies coincide with incentives in the model with describable contingencies. 9