Participation and unbiased pricing in CDS settlement mechanisms. * Ahmad Peivandi † September 01, 2013 Abstract Credit default swaps are insurance contracts on default. Currently, there are about 25 tril- lion USD worth of outstanding CDS contracts. These contracts are settled through a centralized market that has been criticized for underpricing the asset. In this paper, I take a mechanism design approach and characterize robust settlement mechanisms that deliver an unbiased price for the asset. A second contribution of my paper is a new notion of the core for games of in- complete information. This is particularly relevant here because participation in the settlement mechanism cannot be compelled. * I thank Rakesh Vohra for introducing me to this market and for his encouragement to write this paper. I am indebted to Jeffrey Ely for our various valuable conversations. I have received helpful comments from Eddie Dekel and other participants in the CET student seminars. † Northwestern University, Economics Department. Email: [email protected]. 1
Transcript
Participation and unbiased pricing in CDS settlementmechanisms.∗
Ahmad Peivandi †
September 01, 2013
Abstract
Credit default swaps are insurance contracts on default. Currently, there are about 25 tril-lion USD worth of outstanding CDS contracts. These contracts are settled through a centralizedmarket that has been criticized for underpricing the asset. In this paper, I take a mechanismdesign approach and characterize robust settlement mechanisms that deliver an unbiased pricefor the asset. A second contribution of my paper is a new notion of the core for games of in-complete information. This is particularly relevant here because participation in the settlementmechanism cannot be compelled.
∗I thank Rakesh Vohra for introducing me to this market and for his encouragement to write this paper. I amindebted to Jeffrey Ely for our various valuable conversations. I have received helpful comments from Eddie Dekeland other participants in the CET student seminars.
1 IntroductionA Credit Default Swap (CDS) is a contract between two agents whose payoff depends on whethersome third party will default on a bond. In effect, one of the agents in a CDS, called the protectionseller, is insuring the other agent, called the protection buyer, against a third party’s inability topay back a bond. The protection buyer pays a fee for an agreed period of time to the protectionseller and in return receives loss compensation on a reference entity when a credit event occurs.The first modern CDS contract was issued by J.P. Morgan in 1994. Since then, the CDS markethas grown enormously. According to data from the International Swaps and Derivatives DealersAssociation (ISDA), the notional amount of outstanding CDSs was 25.5 trillion USD in early 2011.
To understand how this market works, imagine a reference entity, issuer of the bond, has gonebankrupt. Each CDS contract corresponds to a reference entity’s bond. If AIG, the protectionseller, has issued 10 CDS contracts on the reference entity’s bonds, AIG has to compensate theprotection buyers for their loss on 10 bonds. For instance, if the recovery rate of the defaultedbond was known to be 10 percent and the face value of the bond is $100, AIG has to pay theremaining 90 percent for each contract, which is $9000. The corresponding CDS contracts can besettled via physical settlement or cash settlement. In the case of cash settlement, the protectionseller pays the face value minus the value of the defaulted bond to the protection buyer. In the caseof physical settlement, the protection buyer hands the defaulted bond to the protection seller andreceives the face value of the bond. Physical settlement does not require a price and protectionbuyer’s receive full insurance under physical settlement.
While physical settlement is the natural solution, it is often impossible to physically settle allcontracts. First, in most cases the number of outstanding CDS contracts is more than the numberof bonds.1 If physical settlement were the only way to settle CDS contracts, it would constrainthe number of CDS contracts to the quantity of outstanding bonds on the reference asset.2 Sec-ond, even if protection buyers could purchase the defaulted bonds for physical settlement, doingso would artificially raise the price of the defaulted bond. For these reasons, an alternative way ofsettling the contracts by cash transfer has emerged. The challenge for cash settlement is to identifya value for the defaulted bond. Currently, CDS contracts are settled by a mixture of physical set-tlement and cash settlement.
1As stated in Summe and Mengle (2011) at the time of Delphi Corporation’s bankruptcy it is estimated that therewere $28 billion in CDSs outstanding but only $2 billion in defaulted bond bonds.
2If short selling was facilitated in this market, in physical settlement agents instead of handing the defaulted bondthe protection buyer would short sell the defaulted bond to the protection seller. Since defaulted bonds are traded overthe counter short selling the defaulted bonds is impossible or hard.
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To determine the quantity of contracts to be settled physically or by cash transfer as well asa price for cash settlement, the ISDA introduced a two stage mechanism. In the first stage of themechanism, only agents with CDS contracts participate. This first stage determines the number ofdefaulted bonds to be bought or sold in the second stage of the mechanism and also a price capor floor. In the second stage, a uniform price auction determines a price for the defaulted bond.As of 2009, all CDS contracts are pegged to the value of the defaulted bond determined by thismechanism, unless both protection buyer and protection seller choose to opt out. The ISDA arguesthat requiring all parties to a CDS to be bound by the results of the mechanism ensures certainty,consistency, enhanced transparency, and liquidity. Regarding this requirement, ISDA states ontheir website:
”This is a major milestone in the ongoing refinement of practices and processes for the effi-cient, liquid and transparent conduct of the CDS business, said Robert Pickel, Executive Directorand Chief Executive Officer, ISDA. Hardwiring is central to the many improvements ISDA and theindustry are making to the CDS contract to further ensure that infrastructure and standards fortransacting these important risk management instruments are straightforward, secure and widelyimplemented.”3
The mechanism used by the ISDA has been the subject of criticism. Chernove et al. (2013)have observed that the defaulted bonds in this mechanism are underpriced in the vast majority ofauctions. Underpricing implies that the protection buyer cannot fully insure against the risk ofdefault by the issuer of the bond. This is unpleasant for two reasons: (i) this comes at the cost ofefficiency loss. In an efficient allocation, risk neutral agents (protection sellers) should provide fullinsurance for risk averse agents (protection buyers) against the default risk and (ii) mispricing im-plies that physical settlement and cash settlement have different payoffs, therefore, it would createuncertainty about the payoff of CDS contracts.
The goal of any design should be to settle contracts with unbiased prices. A settlement mech-anism is unbiased if the cash settlement price is equal to the value of the defaulted bond or if anagents’ payoff is equal to the payoff from physical settlement of all contracts. In this paper, I takea mechanism design approach and look for a settlement mechanism that is unbiased. Moreover, Iwant the mechanism to satisfy four important properties which I enumerate:
1. Ex-post incentive compatible, which means that the mechanism is incentive compatible forall possible agents’ beliefs.
3http://www.isda.org/press/press031209.html
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2. Weakly budget balanced, which means that the designer does not have to incur a cost toexecute the settlement mechanism.
3. Context-free, which means that the mechanism is anonymous and robust with respect to thenetwork.
4. Robust with respect to agents’ participation decisions.
The revelation principle implies that, without loss of generality, one can restrict attention directsettlement mechanisms that are incentive compatible, see Myerson (1981). The first property isa robustness property against agents’ beliefs. Since there is cash transfer in the mechanism thesecond property is also standard. Thirdly, a mechanism is anonymous if the prices and quantitiesin two isomorphic networks are the same for corresponding agents. This condition ensures thatthe designer does not discriminate between agents based on their identities. It is robust with re-spect to the network if the price and quantities are the same in two networks where agents have netequal number of contracts. The current mechanism satisfies this property. The rationale for thisproperty is lowering the systemic risk and transaction costs. For more discussion, see section 5.3.3.
Property four is important since this contract does not compel agents to participate in the ISDAsettlement mechanism. Agents may choose to settle some of their contracts outside of the settle-ment mechanism. I define participation-choice to describe the agents’ decision about how theyparticipate in the central mechanism as well as how they settle contracts outside of the mechanism.Participation-choices should satisfy two important criteria. First, when a group of agents makea decision about their participation, it should benefit them. Second, these participation decisionsshould be self-confirming. This means that agents update their beliefs about other agents’ typeswhen other agents join a coalition4, and given the updated beliefs, agents choose to exit when thereis a benefit from doing so. The robustness property listed above is a strong one, which requires themechanism to be unbiased in all possible agents’ participation-choices. The main theoretical inno-vation of this paper is to model how agents choose their level of participation in the mechanism.For more discussion, see sections 5.2.1 and 5.2.3.
A mechanism that sets the cash settlement price equal to the expected value of the defaultedbond conditional on the designer’s information and sets a constant cash settlement quantity5 satis-fies these properties. I call this mechanism the posted price mechanism. I show that all mechanismswith properties 1-4 listed above are payoff equivalent to a posted price mechanism.
4A coalition is a set of agents who choose to settle some of their contracts outside of the settlement mechanism.5Cash settlement quantity may vary across agents but does not depend on reported agents bids.
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As discussed, participation in this mechanism is voluntary. No settlement mechanism canenforce participation. Choosing not to fully participate harms the transparency of the ISDA settle-ment procedure and also drives away liquidity from the auction. This is important because liquidityand transparency were among the main reasons offered for a central mechanism to settle CDSs inthe first place. I show that the posted price mechanism is almost surely the unique mechanism thatensures full participation and leaves no incentives for agents to sell their contracts to other agents.
One may object to posted price mechanisms on the basis of practical implementability. This isbecause there is no clear algorithm that would map publicly available information to a number forthe value of the defaulted bond. Also, if the decision of setting the price is given to the designer,it may give too much power to the designer. If this objection is valid, my result can be thought ofas an impossibility result. This negative result provides a benchmark to design non-trivial mech-anisms. Context-freeness is the only expendable property in this list. I propose an example thatsatisfies properties one, two, and four but violates property three. Moreover, it guarantees that allagents participate with all of their contracts. This mechanism settles all contracts by cash settle-ment. The cash settlement price for the contracts that two agents have is determined only by thereport of other agents.
I extend the characterization result in two ways. First, I consider the case where agents cansell their CDS contracts to other agents prior to the settlement mechanism but after they learn theirprivate signals. Therefore, they can choose their level of participation by either selling their CDScontracts or settling their contracts outside of the mechanism. I provide a richer model for agents’participation. I show the only mechanism that satisfies ex-post incentive compatibility, weak bud-get balancedness, anonymity, and robustness with respect to agents’ decisions about participationis a posted price mechanism. Second, I generalize the notion of unbiasedness. I consider settle-ment mechanisms for which, from an ex-ante point of view, the payoff of each agent is equal to hispayoff from cash settlement with some known price. This property ensures that ex-ante payoff ofagents is proportional to the net number of contracts that they have. This is relevant here becauseCDS contracts are homogenous. I show the characterizations results hold if one replace unbiased-ness with this property.
The rest of the paper is organized as follows. In section 2, I review the current literature. Anexample is provided in section 3 to illustrate the main theoretical contribution of the paper. Theenvironment model is in section 4. Section 5 introduces important properties for the settlementmechanisms. In section 6, I present the results. In section 7, I provide extensions for the resultsand an example of a mechanism that violates the context-freeness property. Concluding remarksare in section 8
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2 Related literatureTo the best of my knowledge, there are five papers written about CDS contract settlement mech-anism, all of which study the current settlement mechanism in use. Gupta and Sundaram (2012)observe that there is a price bias for auctions held in the 2008-2012 period. Similarly, Helwege etal. (2009) compare the mechanism price to the pre and post auction prices of the defaulted bond ina sample of early ten auctions. They find no mispricing in their sample. Coudert and Gex (2010)study the settlement procedure for a number of cases. Their empirical study also reveals a pricebias in the auction. Du and Zhu (2010) develop a theoretical model. Taking the first stage of theauction as exogenous, they show how the winner’s curse results in a price bias in the second period.Abstracting away from an important financial friction, namely the inability to short sell, they pro-pose a double auction which correctly prices the defaulted bond. Their mechanism is essentiallythe physical settlement of all CDS contracts. Chernove et al. (2013) document the same price biasas in Gupta and Sundaram (2012). Taking into account multiple financial frictions in the market,they solve for equilibria of the two stage auction, assuming that agents have no private informationabout the value of the defaulted bond. My paper is the only paper that takes a mechanism designapproach. I look for mechanisms that satisfy the attractive properties enumerated in the in theintroduction.
A second innovation of my paper is a new notion of the core of a game of incomplete informa-tion. I discuss its advantages relative to Myerson (2007), Liu et al. (2012), and Dutta and Vohra(2005). These papers are among the recent papers in the literature. Forges et al. (2002) has asurvey of the older literature. For a detailed discussion see section 5.2.3.
3 Leading ExampleTo illustrate the main theoretical contribution of the paper, I use the following example. The readermay skip reading this example and start from section 4.
Leading Example: There are three agents, 1, 2, and 3. There is a bond with a face value of100. Assume the issuer of the bond has defaulted and the value of the defaulted bond is E[v|s],where s = (s1, s2, s3) is the agents’ signal profile. Agent 1 is a protection seller and agents 2 and 3are protection buyers. Agents 2 and 3 each may have 10 CDS contracts with the protection seller.These homogeneous CDS contracts are on the bond. There are three possible cases (see Fig 1):
1. Agents 2 and 3 have 10 CDS contracts with agent 1.
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2. Agent 2 has 10 CDS contracts with agent 1, and agent 3 has no CDS contracts.
3. Agent 3 has 10 CDS contracts with agent 1, and agent 2 has no CDS contracts.
rAgent 1@@@R
��� rr Agent 2
1010
Agent 3
Case 1
rAgent 1@@@Rr10
Agent 2
Case 2Figure 1: an arrow from agent i to j means that they have contracts and i is a protection seller.
rAgent 1���r 10
Agent 3
Case 3
Denote the number of CDS contracts that agent i has in case j by n ji . Assume n j
i > 0 if agenti is a protection buyer in case j, n j
i < 0 if he is a protection seller, and n ji = 0 if he does not have
any CDS contract. For example, n11 = −20 and n1
2 = n13 = 10.
These contracts are settled by either physical settlement or by cash settlement. In the caseof physical settlement the protection buyer hands in the defaulted bond to the protection sellerand in return receives $100. Therefore, the protection buyer’s payoff from the physical settlementof 1 contract is 100 − E[v|s], and the protection seller’s payoff from the physical settlement is−(100 − E[v|s]). In the case of cash settlement, the protection seller pays the loss to the protectionbuyer(s) in the form of monetary transfer. Therefore, if p is a price for the defaulted bond, theprotection seller pays 100 − p to the protection buyer. If q j
i is the number of agent i’s contractsthat are settled through cash settlement and p j
i is the cash settlement price, agent i’s payoff is asfollows:
(n ji − q j
i )(100 − E[v|s]) + q ji (100 − p j
i ).
Assume agents’ signals about the value of the defaulted bond is either 0 or 1, si ∈ {0, 1} fori ∈ {1, 2, 3}. Assume signals are independently distributed and si = 1 with probability 1
2 . Theexpected value of the defaulted bond conditional on signals is as follows:
E[v|s] = 21(2s1 + s2 + s3).
Assume agents 1 and 2 each possess nine defaulted bonds. Therefore, some of the contractsmust be settled through cash settlement in all cases.
I describe a direct settlement mechanism. A description of a mechanism is a price and a quan-tity function for each agent in each network. The quantity is the number of CDS contracts that are
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settled by cash settlement and the price is the cash settlement price. Let q ji and p j
i denote the quan-tity of cash settlement and cash settlement price for agent i in network j, respectively. Considerthe following settlement mechanism:
One can check that these prices and quantities guarantee ex-post incentive compatibility. More-over, the following holds:
∀s ∈ {0, 1}3 and ∀ j ∈ {1, 2, 3} :3∑
i=1
q ji (s1, s2, s3) = 0, (1)
∀s ∈ {0, 1}3 and ∀ j ∈ {1, 2, 3} :3∑
j=1
q ji (s1, s2, s3)(100 − p j
i (s1, s2, s3)) = 0. (2)
Equation (1) is market clearing condition. Note that q ji (100 − p j
i ) is the cash transfer that agent ireceives in network j, therefore, equation (2) is the budget balanced condition. In addition to theseproperties, if u j
i (s) is agent i’s payoff from the settlement mechanism in case j, the following holds:
Es[uji (s)] = Es[n
ji (100 − E[v|s])].
This condition is called unbiased pricing. It means, from an ex-ante point of view, all contracts aresettled by physical settlement or by cash settlement with the correct price, i.e., p j
i (s) = E[v|s]. Thismechanism is not context-free, since it violates anonymity. This is because the settlement in case2 and 3 are different.
I study agents’ incentives to participate in the settlement mechanism when an arbitrary groupof agents can form coalitions and settle some of their contracts with an arbitrary blocking mecha-nism. As an illustration, consider the settlement mechanism which I described above. I model twocases: full information case and incomplete information case.
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In the full information case, agents observe each other’s signals. Agents 1 and 2 may choose tosettle all of their contracts outside of the mechanism using another mechanism, called the blockingmechanism (see Fig 2). Consider a blocking mechanism in which 6 contracts are settled physicallyand 4 contracts are settled by cash settlement, where the cash settlement prices for agents oneand three are 48 and 49, respectively. The positive spread, the fact that 49 > 48, guarantees apositive payoff for the blocking mechanism designer. Set s3 = 1 and consider a game whoseplayers are agents one and two. In this game, players, after observing the signals, simultaneouslychoose to exit or stay. If both of them choose to exit, then their payoff is their payoff from theblocking mechanism plus the payoff from the settlement mechanism in case 3. If at least one ofthem chooses to stay, the coalition is not formed. In this case, their payoff is only the payoff fromthe settlement mechanism when all three agents are present. There is a Nash Equilibrium in whichagents one and two both choose the following strategy: exit when the signal is zero, and stay whenthe signal is one. In other words, if ue
i is agent i’s payoff when both agents choose to exit, thefollowing inequalities hold:6
u1(0, 0, 1) ≤ ue1(0, 0, 1), u1(1, 0, 1) ≥ ue
1(1, 0, 1), (3)
u2(0, 0, 1) ≤ ue2(0, 0, 1), u2(0, 1, 1) ≥ ue
2(0, 1, 1). (4)
I interpret equations (3) and (4). Assume signal of agent 3 is 1. Equation (3) means that whenthe signal of agent 2 is 0, agent 1 weakly prefers the exit option when his signal is 0 and he prefersthe stay option if his signal is 1. Equation (4) means that when the signal of agent 1 is 0, agent 2weakly prefers the exist option when his signal is 0, and he weakly prefers the stay option if hissignal is 1.
rAgent 1@@@R
��� rr Agent 2Agent 3
1010
Case 1Figure 2
-
rAgent 1�
��rAgent 3
10
Case 3
+
rAgent 1@@@Rr10
Agent 2
Blocking Mechanism
In the incomplete information case, I consider a block by agents 1 and 3 (see Fig 3). In thisblocking mechanism, 7 contracts are settled physically and 3 contracts are settled by cash set-tlement. The cash settlement prices for agents one and three are 30 and 38.5, respectively. The
Therefore, there exists a Bayesian Nash Equilibrium in which agents 1 and 3 choose the exit optionwhen their signals are 0. In this example, when agents 1 and 3 visit the blocking mechanism, i.e.,when (s1, s3) = (0, 0), the blocking designer’s payoff is 3(38.5 − 30).
rAgent 1@@@R
��� rr Agent 2Agent 3
Case 1Figure 3
-
rAgent 1�
��rAgent 3
Blocking Mechanism
+
rAgent 1@@@RrAgent 2
Case 2
Given this model of agents’ participations, in this paper, I answer the following two questions.First, which settlement mechanism ensures all agents to participate with all of their contracts andis also unbiased, budget balanced, and context-free? Second, if we allow agents to settle a numberof their contracts with some blocking mechanisms and take into account agents’ payoff from block-ing mechanisms, which settlement mechanism is unbiased, budget balanced, and context-free? AsI will show, the answer to both questions is a posted price mechanism.
4 PreliminariesWithout loss of generality, I assume the face value of the defaulted bond is 100. Each CDScontract has a protection buyer and a protection seller. In case of a default, the protection buyershould be compensated the loss on the reference asset (bond) by the protection seller. These CDScontracts are homogeneous and each of them corresponds to one bond. I assume that the defaulthas happened, and I consider the contract settlement problem. Let K be the set of all agents. Theseagents may have CDS contracts on the bond between each other. A contract matrix specifies thenumber of contracts that pair of agents have. In contract matrix N = [ni, j], agents i, j ∈ K have netni, j contracts. Assume ni, j > 0 if j is a protection seller and i is a protection buyer, ni, j = 0 if they
7See the appendix for the calculations.
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do not have any CDS contracts, and ni, j < 0 if i is the protection buyer.8 Throughout this paper, Iuse the words network and contract matrix interchangeably.
Let K(N) be the set of agents who have some CDS contracts in network N and ni be the netnumber of contracts that agent i ∈ K has, formally,
K(N) = {i ∈ K|ni, j , 0 for some j ∈ K} and ni =∑
j∈K(N)
ni, j.
Each agent has a number of defaulted bonds; assume agent i has bi ≥ 0 defaulted bonds. Eachagent has a private signal about the value of the defaulted bond. Agent i’s signal is drawn fromS i where S i is a finite subset of real numbers. Given s ∈
∏i∈K S i, a profile of agents’ signals,
the value of the defaulted bond is E[v|s]. Let µ be the agents’ prior about the signals; µ(s) is theprobability of observing the signal profile s. If A ⊆ K is a subset of agents, set S A =
∏i∈A S i.
Given B ⊆ A ⊆ K and s ∈ S A, let πA(s) ∈ S B be the projection of s on its’ B elements. I assumeE[v|s] is non-decreasing in agents’ signals. That is, given s−i ∈ S K\{i} and si, s′i ∈ S i, the followingholds:
si > s′i ⇒ E[v|s−i, si] ≥ E[v|s−i, s′i].
I assume that agents know the exact functional form of the value function. But the designer hasa belief about the functional form and agents’ priors, µ. Let κ be the support of designer’s belief.The support consists of pairs of functional form and agent’s priors, in other words, elements of κare of the form of (µ, E[v|s]). Let ρ be the designer’s prior about the pair of agents’ prior and valuefunction. I use Eρ to refer to the expected value symbol given the designer’s information.
These CDS contracts are settled by either physical settlement or by cash settlement. In thecase of physical settlement, the protection buyer hands in the defaulted bond to the protection sellerand in return receives 100. Therefore, the protection buyer’s payoff from the physical settlementof 1 contract is 100 − E[v|s], and the protection seller’s payoff from the physical settlement is−(100 − E[v|s]). In the case of cash settlement, the protection seller pays the loss to the protectionbuyers. Therefore, if pi is the price of the defaulted bond, and q j
i is the number of agent i’s contractsthat are settled through cash settlement, agent’s payoff at signal profile s ∈ S K is as follows:9
(ni − qi)(100 − E[v|s]) + qi(100 − pi).
One can rewrite the payoff of agent i as follows:
ni(100 − E[v|s]) + qi(E[v|s] − pi).8Note that ni, j + n j,i = 0.9Agent i gives/receives ni − qi of his defaulted bonds and receives/pays the face value of the bond, in addition he
receives/pays his loss for the rest of the contracts.
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Where the first term is his payoff if all of the contracts are physically settled or if the price isequal to the value of the defaulted bond. The second term can be thought of as the bias. Ingeneral, a settlement mechanism is a reallocation of the defaulted bonds and monetary transfer.Note that combinations of physical settlement and cash settlement can generate any allocation ofthe defaulted bonds and monetary transfer.
An agent is short selling if he ends up with net a negative number of defaulted bonds afterthe settlement, that is: bi < ni − qi. As I will show, the problem is trivial if the market facilitatesshort selling. However, short selling in this market is generally hard or impossible. Therefore, anysettlement should satisfy the no short sell constraint, which is qi ≥ ni − bi for all i ∈ K.
5 Settlement Mechanisms
5.1 Description of a MechanismIn this environment, a direct settlement mechanism takes the network and the profile of reportedsignals as inputs and returns a cash settlement quantity and a cash settlement price for each agent.A direct mechanism consists of functions qN
i : S K → R and pNi : S K → R for all agents i ∈ K and
network N. The cash settlement quantity is qNi , and pN
i is the cash settlement price for agent i inthe network N. Let pN = (pN
i )i∈K and qN = (qNi )i∈K be the profile of price and quantity functions
when the network is N and also p = (pN) and q = (qN) be the price and quantity profiles. Notethat I allow agents to have different cash settlement prices; in other words, I am not restricting topN
i = pNj for all i, j ∈ K. Therefore, any reallocation of money and the defaulted bonds can be
generated by cash settlement and physical settlement. Number of defaulted bonds that are used forphysical settlement must clear itself, formally, for all networks N and s ∈ S K:∑
i∈K
(ni − qNi (s)) = 0.
This is equivalent to∑
i∈K qNi (s) = 0. This mechanism is ex-post incentive compatible if for all
networks N, i ∈ K, and s = (si, s−i) & s′ = (s′i , s−i) ∈ S K , the following holds:
(ni − qNi (s)(100− E[v|s]) + qN
i (s)(100− pNi (s)) ≥ (ni − qN
i (s′))(100− E[v|s]) + qNi (s′)(100− pN
i (s′)).
This means that agent i with private information si should not find it profitable to misreport hissignal as s′i , when all other agents’ signal profile is s−i.
I use the notation (p, q, u) for a settlement mechanism with price, quantity, and payoff functionspN
i , qNi , and uN
i for all networks N and agents i ∈ K. Agent i’s payoff in the network N when allagents are reporting their signals truthfully is as follows:
uNi (s) = ni(100 − E[v|s]) + qN
i (s)(E[v|s]) − pNi (s)).
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Note that the cash settlement part of an agent’s payoff, qNi (s)(E[v|s]) − pN
i (s)), is a monetarytransfer to the agent. The mechanism is ex-post budget balanced if for all networks N and s ∈ S K ,the equality
∑i∈K qN
i (s)(100 − pNi (s)) = 0 holds. It is ex-post weakly budget balanced if for all
networks N and s ∈ S K , the inequality∑
i∈K qNi (s)(100 − pN
i (s)) ≤ 0 holds. It is ex-ante budgetbalanced if, Eρ[
∑i∈K qN
i (s)(100 − pNi (s))] = 0 for all networks N. I define ex-ante weakly budget
balanced mechanisms similarly. Since for each network N sum of nis and qis are zero for alls ∈ S K ,
∑i∈K qN
i (s)(100 − pNi (s)) =
∑i∈K ui(s). This implies that the mechanism is ex-ante weakly
budget balanced if, Eρ[∑
i∈K uNi (s)] ≤ 0 for all network N. I restrict attention to ex-post incentive
compatible and ex-ante weakly budget balanced settlement mechanisms. A mechanism has noshort sell if qi(s) ≤ ni − bi for all i ∈ K and s ∈ S K .
5.2 Participation IncentivesAn agent who does not have any CDS contract is not obligated to participate in the settlementmechanism; he participates if there is a positive payoff. This motivates the following defini-tion: a mechanism is ex-post individually rational for agents without contracts if, for all net-works N, all signal profiles s ∈ S K , all agents i ∈ K that satisfy ni, j = 0 ∀ j ∈ K, the in-equality qN
i (s)(100 − E[v|s]) ≥ 0 holds. It is interim individually rational if the inequalityEs−i[q
Ni (s)(100 − E[v|s])] ≥ 0 holds.
I formally model the participation decision of agents who have CDS contracts. In standardmechanism design, agents can choose whether or not to participate in the mechanism. They par-ticipate when they have a nonnegative payoff from participating in the mechanism. Participationin this environment is different for an important reason. Agents with CDS contracts are requiredto participate by default, however, if both parties of a CDS contract agree, they can settle some oftheir contracts with another mechanism. In this environment, agents’ outside options are no longerexogenous; rather they depend on their signals as well as other agents’ signals. In other words, ifan agent agrees to settle a CDS contract though another mechanism, it reveals information abouthis own private signal. I do not assume the number of contracts that a pair of agents have is privateinformation; rather, agents are legally allowed to not bring a number of their contracts to the set-tlement mechanism, if parties all parties of these contracts agree.
Because of these bilateral decisions that agents can make prior to participating in the mech-anism about the number of contracts, the designer may face contract matrices different from theoriginal network of contracts. When the contract matrix is N, if a group of agents choose to settlesome of their contracts outside of the settlement mechanism, the designer faces a new contractmatrix, namely M. In this case M is a reduction of N. Formally, M = [mi, j] is a reduction of
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N = [ni, j] if, for all i, j ∈ K, the inequality |mi, j| ≤ |ni, j| holds. I use the notation M ≺ N, if M is areduction of N. Let A be the set of all agents who chose to settle some of their contracts outside ofthe settlement mechanism. Note that A = K(M − N) where M − N is a contract matrix in whichagents i and j have mi, j − ni. j contracts. Actions of agents in A have reduced network N to networkM.
A blocking mechanism can be viewed as a settlement mechanism when the set of agents isA and the network of contracts is N − M. The main differences are that it does not have to bebudget balanced and it does not have to clear the number of defaulted bonds used. A blockingmechanism has an important role, it is where all contracts that were not brought to the settlementmechanism are settled. I present two models for the blocking: (i) complete information case and(ii) incomplete information case.
5.2.1 Complete Information Case
Agents in A for a subset of their types block the settlement mechanism and reduce the networkfrom N to M if there exists a blocking mechanism (p′, q′, u′), sK\A ∈
∏i∈K\A S i a profile of types
for agents that are not in A, and prescribed subset of types S ′i ⊆ S i for agents in A, such that thefollowing holds:
1. For all s ∈ S K and i ∈ A such that πA(s) ∈ Π j∈AS j, the following inequality holds:
uNi (s) ≤ u′i(s) + uM
i (s). (6)
Note that u′i(s) = (ni − mi − q′i(s))E[v|s] + q′i(s)(100 − p′i(s)) is agent i’s payoff from theblocking mechanism. Agents in A join the coalition when their types are in the prescribedsubset of types. Inequality (6) means that if all signals of agents A are in the prescribed sets,then the payoff of agent i ∈ A from the settlement mechanism with network N is not largerthan his payoff from the blocking mechanism plus the payoff from the settlement mechanismwith network M. This gives agent i incentives to join the coalition when all blocking agents’signals are in the prescribed sets.
2. For all s ∈ S K such that πA\{i}(s) ∈∏
j∈A\{i} S ′j, π{i}(s) ∈ S i \ S ′i , and πK(N)\A(s) = sK(N)\A, thefollowing inequality holds:
uNi (s) ≥ u′i(s) + uM
i (s). (7)
Inequality (7) means that if agent i’s signal is not in S ′i and the signal of all other agentsin A are in the prescribed sets, then agent i’s payoff from the settlement mechanism with
14
network N is not smaller than his utility from the blocking mechanism plus his payoff fromthe settlement mechanism with network M.
3. The blocking designer has a positive payoff. Formally, the following inequality must hold:
E[−∑i∈A
u′i(s)|s ∈∏j∈A
S ′j × {sK\A}] > 0. (8)
Inequalities (6) and (7) mean that agents in A, for a subset of their private signals, may form acoalition and settle some of their contracts with the blocking mechanism. Agents in the coalition,A, choose between (pN , qN , uN) and (pM, qM, uM) plus the blocking mechanism. Consider a gamein which agents in A choose to exit the mechanism or stay. The block is formed only when allagents choose to exit. Assume the signal profile of agents that are not in A is sK\A, if agents choosethe following strategy: exit only if the type is in the prescribed set; these inequalities guarantee thatupon learning everyone’s type, no agent would regret his decision. The mechanism is unraveledif blocking exists.
To understand inequality (8), think of the blocking designer as an agent. Note that in generalthe blocking mechanism does not have to balance the budget or clear the number of defaultedbonds that are used for physical settlement. Since there may be a surplus or a deficit in (i) mon-etary transfer and (ii) the number of defaulted bonds, the blocking designer’s payoff may not bezero. Inequality (8) means that the blocking designer’s expected payoff conditional on the eventthat the block is formed must be positive. The first term that appears in the summation is the block-ing designer’s payoff from defaulted bonds and the second term is his payoff from the monetarytransfer.
Proposition 5.1. Let M and N be a pair of contract matrices where M is a reduction of N. Let Abe defined as in the definition of blocking and (p, q, u) be a settlement mechanism. If the sum ofagents’ payoffs that are in A is greater when the network is N compared to that of network M, inother words inequality (17) holds, then there is no complete information unraveling for which N isreduced to M by agents in A. Moreover, if pM
i , qMi do not depend on s, then the inequality is also a
necessary condition. ∑i∈A
uNi (s) ≥
∑i∈A
uMi (s) for all s ∈
∏i∈K
S i (9)
Proof. The sufficient condition’s proof is by contradiction. Assume there is a blocking mechanism(p′, q′, u′). If one adds up inequality (6) in the definition of a blocking mechanism for all i ∈ A,then for all s ∈ S K such that πK\A(s) = sK\A, the following holds:∑
i∈A
uNi (s) ≤
∑i∈A
uMi (s) + u
′
i(s). (10)
15
Inequalities (17) and (10) imply the following inequality:
E[∑
u′i(s)|s ∈∏j∈A
S ′j × {sK\A}] < 0. (11)
Hence, the following holds:∑i∈A
E[uNi (s)|s ∈
∏j∈A
S ′j × {sK\A}] <∑i∈A
E[uMi (s)|s ∈
∏j∈A
S ′j × {sK\A}]. (12)
Therefore, for some s ∈ S K ,∑
i∈A uNi (s) <
∑i∈A uM
i (s). This contradicts with the assumption in theproposition.
To establish the necessary part of the proposition, let (q, p, u) be a settlement mechanism whichdoes not satisfy 17, i.e., for some s∗ = (s∗A, s
∗K\A) ∈ S K:∑
i∈A
uNi (s∗) <
∑i∈A
uMi (s∗). (13)
I construct the blocking mechanism (q′, p′, u′). For all i ∈ A and sA ∈ S A, set q′i(sA) = qNi (s∗) −
qMi (s∗) and p′i(sA) = p′i where p′i is the unique solution to the following equation:
uNi (s∗) = uM(s∗) + (ni − mi − q′i(sA))(100 − E[v|s∗]) + q′i(sA)(100 − p′i).
This equation has a unique solution since the right hand side is linear in p′i . Set S ′i = {s∗i } forall i ∈ A. The mechanism (p′, q′, u′) is ex-post incentive compatible, since for all i ∈ A, p′i andq′i do not depend on si. Inequality (6) is satisfied by construction. To check inequality (7), leti ∈ A and s′′i , s′i . Let s0 ∈ S be such that π{i}(s0) = s′′i and for all j ∈ A \ {i}, π{ j}(s0) = s′j andπK\A(s0) = s′K\A. If (s′′i , sK\{i}) ∈ S K and s j = s′j for all j ∈ I & j , i, incentive compatibility of thesettlement mechanism and construction of the (p′, q′, u′) mechanism imply:
uNi (s0) ≥ uN
i (s′) + qNi (s′)(E[v|s0] − E[v|s′]) = uM
i (s′) + u′i(s′) + qNi (s′)(E[v|s0] − E[v|s′])
≥ uMi (s0) + u′i(s0) + (−qM
i (s′) − q′i(s′) + qNi (s′))(E[v|s0] − E[v|s′]) = uM
i (s0) + u′i(s0)
Because of (13) the blocking mechanism designer has a positive payoff at signal profile s∗. �
A settlement mechanism (p, q, u) is full information unravel-proof if for any pair of contractmatrices M and N and subset of agents A ⊆ K, where agents in A reduced N to M, agents in Acannot form a block for a subset of their types. I show a corollary of proposition 5.1. The followingcorollary ensures that the set of unravel-proof mechanisms is not empty.
16
Corollary 5.2. Settlement mechanism (p, q, u) for which pNi = Eρ[v] and qN
i = ni is unravel-proof.This is a mechanism that settles all of the contracts by physical settlement.
Proof. Since pNi and qN
i do not depend on s, all I need to show is that the settlement mechanismsatisfies the sufficient conditions in proposition 5.1. Given N and a reduced contract matrix M:∑
i∈A
uNi (s) =
∑i∈A
ni(100 − E[v]) = (100 − E[v])∑i∈A
ni =
(100 − E[v])(∑i∈K
ni −∑
i∈K\A
ni) = −(100 − E[v])∑
i∈K\A
ni.
By construction of A, if i < A then ni = mi, therefore the following holds:
−(100 − E[v])∑
i∈K\A
ni = −(100 − E[v])∑
i∈K\A
mi =∑i∈A
uMi (s).
�
The novelty in this paper is that I consider the possibility that the outside option of an agentdepends on the set of other agents who exercise their outside option. This is because the agentswho choose to ‘exit’ may decide to band together and settle some of their contracts among them-selves through a different mechanism. Such a possibility was first considered in the literature oncooperative games, culminating in the notion of the core. The notion of unraveling presented inthis paper is related to the block in matching theory and the block in cooperative game theory.The difference between blocking in matching theory and the notion of unraveling is that in my setup the network is predetermined and only price and quantity of cash settlement is chosen thougha mechanism. Unravel-proofness is a property for a mechanism that is defined over networks butstability is defined over a possible match. Unlike similar concepts in corporate game theory, the no-tion of unravel-proofness can be naturally extended to environments with incomplete information.A generalization of the unravel-proofness notion to environments with incomplete information ispresented in the following section.
5.2.2 Incomplete Information Case
I extend the blocking mechanism definition to environments where agents do not know each other’ssignals but share a prior. When agents make decisions about whether or not to join the blockingmechanism, they update their belief upon observing other agents decisions. An agent’s choice toparticipate in a blocking mechanism reveals that his private signal is in some subset of types. Otheragents take this into account when making their decisions.
17
For all i ∈ A, subsets ∅ , S ′i ⊆ S i are called the prescribed sets. Let event Ei be defined asfollows:
Ei = {s ∈ S K |πA\{i}(s) ∈∏
j∈A\{i}
S ′j , π{i}(s) ∈ S i \ S ′i}.
Define event E asE = {s ∈ S K |πA(s) ∈
∏j∈A
S ′j}.
Note that Ei is the event that the private signal of all agents in A, except for agent i are in theprescribed sets. Event E is the event that private signal of all agents in A are in the prescribed sets.The inequalities in the blocking mechanism definition, inequalities (6), (7), and (8), change to thefollowing inequalities:
Es−i[uNi (s)|E] ≤ Es−i[u
′i(s) + uM
i (s)|E], (14)
Es−i[uNi (s)|Ei] ≥ Es−i[u
′i(s) + uM
i (s)|Ei]. (15)
E[−∑i∈A
u′i(s))|E] > 0. (16)
To interpret inequalities (14) and (15), imagine a game whose players are agents in A. Theseagents, after observing their private signals, choose whether or not to participate in the blockingmechanism. If all of these agents decide to participate in the blocking mechanism, their payoff istheir payoff from the blocking mechanism plus their payoff from the settlement mechanism, whenthe network is M. If some of them decide not to participate in the blocking mechanism, their pay-off is only the payoff from the settlement mechanism, when the network is N. The mechanism isunraveled if this game has a Bayesian Nash Equilibrium in which agents in A for a subset of theirtypes choose the blocking mechanism. With the new definition of a block, unravel-proofness isnaturally redefined. I call this new notion incomplete information unravel-proof.
Proposition 5.3. Let M and N be a pair of contract matrices where M is a reduction of N. Let Abe defined as in the definition of blocking and (p, q, u) be a settlement mechanism. If the expected10
sum of agents’ payoffs that are in A is greater when the network is N compared to that of networkM, in other words inequality (17) holds, then there is no incomplete information uraveling forwhich N is reduced to M by agents in A. Moreover, if pM
i , qMi do not depend on s, then the inequality
is also a necessary condition.
ES K\A[∑i∈A
uNi (s)] ≥ ES K\A[
∑i∈A
uMi (s)] for all s ∈
∏i∈K
S i (17)
Proof. The proof is an adaptation of the proof of 5.1 and hence is omitted. �
10Conditional on the signal profile of agents in A
18
5.2.3 Prior Notions of Core
The notion of unravel-proofness under incomplete information can be interpreted as a stabilitycondition. The set of mechanisms that satisfy the property can be interpreted as the core of theunderlying game of incomplete information. The notion of core has been generalized to gamesof incomplete information in Wilson (1978), Dutta and Vohra (2005), Myerson (2007), and Liu etal. (2012). These notions differ in the way that agents communicate their private signals. Wil-son (1978) considers two extreme cases: (i) all agents in a block share their private informationcompletely (fine core) (ii) where agents share no private information. Dutta and Vohra (2005)and Myerson (2007) consider the blocks for which the decision to join the block comes from aBayesian Nash Equilibrium. Liu et al. (2012) study the implications of common knowledge ofstability of a two sided match when one side of the market has incomplete information about theother side.
My notion of core differs from other notions introduced in the literature in two important ways.First, in the prior notions of core the decision to join the blocking mechanism is after the realizationof the grand mechanism’s allocation. Therefore, the blocking designer takes the allocation ofthe grand mechanism as exogenous. Since my set up is agents with quasilinear preferences andcommon value, no trade theorem implies that no subset of agents should agree to a reallocationonce the contracts are settled, see Milgrom and Stokey (1982). In my notion of the block, agentssimultaneously choose whether they want to participate in the blocking mechanism. The seconddifference is that in my setup blocking mechanism and the settlement mechanism can coexist. Thisis because an agent may choose to partially participate in the settlement mechanism.
5.3 Unbiased MechanismsAs I mentioned in section 2, several authors have criticized the current settlement mechanism inuse for underpricing the bond. I look for mechanisms that overcome this issue. Because of theinformation rent, due to agents’ private information about the value of the defaulted bond, ex-postcorrect pricing is not possible. However, I consider a weaker condition; I look for mechanisms thatare unbiased from the ex-ante perspective. I define unbiasedness in two ways.
5.3.1 Weakly-Unbiased
A weakly-unbiased mechanism is the one that does not misprice the defaulted bond in expectation.Formally, Mechanism (p, q, u) is weakly unbiased if, for all networks N and agents i ∈ K:
E[uNi (s)] = [ni(100 − E[v|s])]. (18)
19
Equation (18) means that from an ex-ante perspective the agents payoff from the settlementmechanism is the same as their payoff from physical settlement of all contracts or cash settlementwith the price equal to the value of the defaulted bond. Note that since both price and quantity maydepend on the signal profile, this condition is not equivalent to E[pN
i (s)] = E[v].
Observation 5.1. If E[uNi (s)] = ni(100 − E[v]), then Eρ[
∑i∈K uN
i (s)] = (∑
i∈K ni)(100 − E[v]) = 0.Therefore, a mechanism is weakly ex-ante budget balanced, if it is weakly-unbiased.
5.3.2 Unbiased
Since agents may settle some of their contracts outside of the settlement mechanism, agents totalpayoff is not only the payoff from the settlement mechanism; rather, it should include the payoff
from the blocking mechanisms. A mechanism is unbiased, if the agents’ total payoff, includingthe payoff from the settlement mechanism and the blocking mechanisms, from an ax-ante perspec-tive, is equal to the agents’ payoff from physical settlement of all contracts or cash settlement withthe correct price.
To formally define unbiased mechanisms, I first define participation-choice. For some net-works, a group of agents may find it profitable to settle some of their contracts with a blockingmechanism. Using the techniques from the definition of unravel-proofness, I allow agents to takethese actions, a mechanism is unbiased, if it is weakly-unbiased regardless of these actions.
Consider an ex-post incentive compatible settlement mechanism, namely (p, q, u), which maynot be unravel-proof. A participation-choice is a collection of sets (PN)N where elements of PN
capture the sub networks that join a coalition if the true network of contract is N. For all networksN, each element of c ∈ PN has a network cN ≺ N, a subset of agents cA ⊆ K(cN), a set of typeprofiles where ct =
∏i∈cA
S ci for all i ∈ cA, where S c
i ⊆ S i, and a blocking mechanism for the cN
network, ( pc, qc, uc). This participation-choice should satisfy three conditions.
1. Let AN(s) be the set of all coalitions that are formed when the signal profile is s ∈ S K .Formally,
AN(s) = {c ∈ AN |πcA(s) ∈ ct}.
It must be that N(s) =∑
c∈AN (s) cN ≺ N. This means that the network which is left aftercoalitions are formed is a reduction of N.
2. Agent i’s payoff from this participation-choice when the network is N is as follows:
uPNi (s) =
∑c∈AN (s):i∈cA
uci (s) + uN−N(s)
i (s).
20
Joining the coalitions for the prescribed types must be a Bayesian Nash Equilibrium. For-mally, for all networks N and c ∈ AN , let events Ec and Ei
c be defined as:
Ec = {s ∈ S K |πAc(s) ∈∏
j∈K(cN )
S cj},
Eic = {s ∈ S K |πAc(s) ∈
∏j∈K(cN )\{i}
S cj , π{i}(s) ∈ S i \ S c
i }.
For all i ∈ cA, the following inequalities should hold:
Es−i[uPNi (s)|Ec] ≥ Es−i[u
PN\{c}i (s)|Ec],
Es−i[uPNi (s)|Ei
c] ≤ Es−i[uPN\{c}i (s)|Ei
c].
3. Given a coalition c ∈ AN , when the signal profile is s ∈ S K , agent i ∈ K(cN) enters the coali-tion c if πcA(s) ∈ ct. The blocking designer’s expected payoff from the blocking mechanismshould be positive conditional on the event that all agents in cA join the blocking mecha-nism. This is similar to inequality (16), which insures that the blocking mechanisms areself-sustaining.
Consider a participation-choice for each functional form E[v|s] and agents’ prior µ. The mecha-nism is unbiased, if for all networks N, all agents i ∈ K, and all participation-choices the followingholds:
Eρ[uPNi (s)] = Eρ[ni(100 − E[v|s])]. (19)
This condition says, from an ex-ante perspective agent i’s total payoff from the blocking mecha-nism and the settlement mechanism for all possible participation-choices is equal to his payoff fromphysical settlement of all contracts. Note that since I allow some contracts to be settled outside ofthe settlement mechanism, the notion of budget balancedness must be modified. The mechanismis weakly budget balanced regardless of agents’ participation-choice if, for all networks N and allparticipation-choices the following holds:
Eρ[∑
i∈K(N)
uN−N(s)i (s)] ≤ 0.
5.3.3 Context-Free Properties
In this section, I introduce and motivate two context-free properties.
21
Consider a settlement mechanism which may not be a direct mechanism. This mechanism isanonymous if the prices and quantities are the same in two isomorphic networks. Here is a formaldefinition. A mechanism is anonymous if for all pair of networks M = [mi, j] and N = [ni, j] thatsatisfy the following condition:
• There exists an onto and one to one mapping f : K → K that satisfies ni, j = m f (i), f ( j) for alli, j ∈ K.
The equalities qNi (s) = qM
f (i)(s) and pNi (s) = pM
f (i)(s) hold for all agents i ∈ K.
I introduce the second property. The current mechanism that is in use only takes the net num-ber of contracts as an input and not the detail of the network of contracts. The rationale for thisproperty is lowering the systemic risk and transaction costs. To elaborate, consider the followingexample. Agent 1 has sold a number of CDSs to agent 2 and agent 2 has sold the same amount ofCDSs to agent 3. If agent 1 is unable to settle these contracts with agent 2, it would harm all agentsin the chain. To get around this issue the clearing house treats agent 2 as an agent without any CDScontract. Also netting results in fewer transactions and hence reduces the transaction cost. Thismotivates the property of robust with respect to network. Here is the formal definition, a mecha-nism is robust with respect to the network if, for all pair of networks M = [mi, j] and N = [ni, j] thatsatisfy
∑j∈K mi, j =
∑j∈K ni, j for all i ∈ K, the following equality holds: ∀i ∈ K, qN
i (s) = qMi (s) and
pNi (s) = pM
j (s).
A mechanism is context-free if it satisfies the property of anonymity and is robust with respectto the network.
6 ResultsI look for mechanisms that satisfy a combination of properties that I have introduced in the previoussection. Before presenting the characterization result, I introduce a class of mechanisms called theposted price mechanisms.
6.1 Posted Price MechanismsA mechanism (p, q, u) is a posted price mechanism if the following is satisfied for prices andquantities:
22
• For all agents i ∈ K and networks N, qNi (s) =
∑j∈K qni, j . Where qn, for all |n| < n, is a real
number that satisfies q−n = −qn.11
• For all agents i ∈ K and networks N, pNi (s) = p for some p ∈ R.
The first condition says that the quantity function is separable and the second condition says thatthe price is constant. In this class of settlement mechanisms prices and quantities do not depend onthe reported signals. If qn = 0 for all n, then the settlement mechanism is the physical settlementof all contracts.
No short sell constraint can be easily imposed on this class of settlement mechanisms. It mustbe that for all networks N and i ∈ K(N),
∑j∈K(N) qni, j ≤ ni − bi.
A posted price mechanism is the one for which the price is equal to the ex-ante value of thedefaulted bond conditional on the designer’s information. In other words, p = Eρ[v].
Proposition 6.1. Any posted price mechanism with a fair price is context-free, unbiased, unravel-proof (both complete information and incomplete information), and ex-post individually rationalfor agents without contracts.
Proof. The only non-trivial property is unravel-proofness. Note that mechanisms in this classsatisfy the sufficient conditions in propositions 5.1 and 5.3. Therefore, they are unravel-proof. �
6.2 CharacterizationsTheorem 6.2. If (p, q, u) is a settlement mechanism that is full information unravel-proof, context-free, weakly-unbiased, and ex-post individually rational for agents without contracts, then almostsurely it is payoff equivalent to a posted price mechanism with a fair price.
Proof. A sketch of the proof is presented, for the complete proof see the appendix. Proof is byinduction. First, I establish the case where there are only two agents. For the case of two agents, Iuse budget balancedness and anonymity properties. To prove the inductive step, I use the followinglemma.
Lemma 6.3. Consider the assumptions and the set up in proposition 5.1, if the settlement mech-anism is unbiased and the mechanism is payoff equivalent to a posted price mechanism when thenetwork is M, then almost surely for all s ∈ S K:∑
i∈A
uNi (s) =
∑i∈A
uMi (s).
11Note that q0 = 0.
23
This lemma connects the mechanism in a network to its reductions. Using this lemma, I com-plete the inductive step by considering three different cases. �
The unravel-proofness property in theorem 1 is the full information notion. The same resultstill holds with the imperfect information model, if I make the following assumption about thedesigner’s belief.
Assumption 6.1. Full Rank Belief Let κ0 ⊂ κ be a zero probability set. Given the functional formE[v|s] and signal si for agent i,
if∑s−i
µ(s−i|si)x(s−i) = 0 ∀(E[v|s], µ) ∈ κ − κ0 then x(s−i) = 0 for all s−i
This assumption resembles the assumption in Cremer and Mclean (1988) but it is different.
Theorem 6.4. Under assumption 6.1, if (p, q, u) is a settlement mechanism that is incompleteinformation unravel-proof, context-free, weakly-unbiased, and interim individually rational foragents without contracts, then almost surely it is payoff equivalent to a posted price mechanismwith a fair price.
Proof. See the appendix for the proof. �
The motivation for unravel-proofness is that participation in the mechanism ensures liquid-ity and transparency of the settlement procedure. However, this property can be replaced with astronger version of the weak-unbiasednss property, namely the unbiasedness property.
Theorem 6.5. Under assumption 6.1, if (p, q, u) is a settlement mechanism that is context-free, un-biased, weakly budget balanced regardless of agents participation choice, and interim individuallyrational for agents without contracts, then almost surely it is payoff equivalent to a posted pricemechanism with a fair price.
Proof. See the appendix for the proof. �
7 Discussion
7.1 A Proposed MechanismAs discussed in the introduction, a posted price mechanism might be hard to be practically imple-mented. Therefore, the results show that in order to design a mechanism that aggregates agents’information, one needs to relax one of these properties (i) unbiasedness or weakly-unbiasedness
24
& unravel-proofness, (ii) context-freeness, and (iii) ex-post incentive compatibility. The obviouscandidate in this list is the context-freeness property. I provide a mechanism that satisfies all theseproperties except for context-freeness.
Here is the description of the mechanism. All agents report an estimate about the value of thedefaulted bond. All contracts are settled by cash settlement. The contracts between two agentsare settled by the price equal to the average of the report of all other agents (excluding these twoagents).
I argue that this simple settlement procedure satisfies (i) ex-post incentive compatibility, (ii)unbiasedness, and (iii) unravel-proofness. Note that since an agent’s report does not affect howhis contracts are settled, they don’t have a profitable incentive to misreport their estimate. On topof that it has been argued in Chernove et al. (2013) that agents’ reputation is important for them.Therefore, a truthful report is sustained in the equilibrium. I show this mechanism is unravel-proof.The price at which contracts that are between agents i, j ∈ K are settled is pi, j =
∑l,i, j E[v|sl]|K|−2 . There-
fore, agent’s i’s payoff in network N is∑
l,i ni,l(100− pi, j). Note that the assumption in proposition5.1 is satisfied with equality. Therefore, this mechanism is unravel-proof.
The only objection to this mechanism is the possibility of collusion. Selling the CDS contractsprior to participating in the mechanism can be thought of as a form of collusion, in the next section,I show even if one drop the notion of robustness with respect to the network, the posted pricemechanism is the only robust mechanism that is unbiased and disincentives agents from sellingtheir contracts.
7.2 Extensions7.2.1 Selling CDS Contracts
Agents are legally allowed to sell some of their CDS contracts prior to participating in the settle-ment mechanism. In this section, I explore the possibility that agents trade their contracts beforethe settlement mechanism, but after they learn their signals. This motivates to change the definitionof unravel-proofness to make sure agents do not have incentives to take the following two actions(i) settling some of their contracts with another mechanism and (ii) selling some of their contracts.I change the definition of network reduction as follows:Formally, M is a reduction of N if there exists a sequence of networks (Mt = [mt
i, j]i, j∈K)t=τt=0 such that
M0 = N, M1, M2,...,Mτ = M and given Mt for 0 ≤ t ≤ τ − 1, Mt+1 satisfies one of the followingtwo cases:
25
1. Contract matrix Mt+1 , Mt is such that for all i, j ∈ K, |mti, j| ≥ |m
t+1i, j | and if mt
i, j , 0, then mti, j
and mt+1i, j have the same sign. In this case, set of agents who took this action are all agents
i ∈ K such that mti, j , mt+1
i, j for some agent j ∈ K .
2. Contract matrix Mt+1 is constructed from Mt by removing some contracts that are betweentwo agents i ∈ K and j1 ∈ K and adding these contracts to contracts between i and anotheragent, j2 ∈ K. It must be that
mt+1i, j1 + mt+1
i, j2 = mti, j1 + mt
i, j2 .
No other contract is removed or added. This is the case where agent j1 buys some of thecontracts that are between j2 and i from agent j2. In this case, agents j1 and j2 took anaction.
The definition of unraveling is similar to the previous case. Let A be the set of agents who (poten-tially) take these actions. These agents block and reduce the network from N to M if there existsa blocking mechanism that satisfies the inequalities in the definition of blocking.12 The blockingmechanism has two roles: (i) it is where all contracts were not brought to the settlement mecha-nism are settled and (ii) it is where agents are compensated for selling their CDS contracts. Withthis modification, unbiasedness and unravel-proofness are naturally redefined.
Theorem 7.1. Given the new definitions, the property of robustness with respect to the network intheorems 6.2, 6.4, and 6.5 can be dropped.
Proof. See the appendix. �
7.2.2 Ex-ante Uniform Price
I propose a generalization for the unbiased and weakly-unbiased properties. A mechanism isweakly ex-ante uniform price if from an ex-ante perspective, CDS contracts have the same payoff.Without this property, different contracts will be settled differently, even from an ex-ante perspec-tive.
Here is a formal definition. A mechanism satisfies weak ex-ante uniform price property if forsome price function p : S → R, all networks N, and all agents i ∈ K:
Eρ[uNi (s)] = Eρ[ni(100 − p(s))].
12For the case of complete information these inequalities are inequality (6), (7), and (8) and for the case of incom-plete information, they are (14), (15), and (16).
26
Ex-ante uniform price property is defined naturally. Given a participation-choice for eachfunctional form and agents prior, agent i’s ex-ante payoff is defined as E[uPN
i (s)], for all s ∈ S K .The mechanism is unbiased, if for all agents i ∈ K, all networks N and all participation plans andsome price function p : S → R, the following holds:
EρuPNi (sK(N))] = Eρ[ni(100 − p(s))]. (20)
Proposition 7.2. Properties of unbiasedness and weakly-unbiasedness can be replaced with ex-ante uniform price and weakly ex-ante uniform price, respectively, in theorems 6.2, 6.4, 6.5, and7.1.
Proof. See the appendix for a proof. �
7.3 Counter ExampleExample 7.1. Assume there are two agents with one contract where agent 1 is a protection buyer.Agent 1 has signals {G1, B1} and agent two has signals {G2, B2}, signals are independent with equalprobabilities. Assume E[v|B1, B2] = 1, E[V |B1,G2] = 2, E[v|B2,G1] = 3 and E[v|G1,G2] = 4.Consider the following price and quantities:
This mechanism is ex-post incentive compatible, unbiased, unravel-proof, and weakly budget bal-anced for this case. This mechanism violates property of anonymity. Since if these prices andquantities are used when agent 1 is protection seller it would violate incentive compatibility.
8 ConclusionI took a mechanism design approach to address the design of CDS contract settlement problem.The design would have been trivial if short selling was possible, because one could settle all con-tracts physically. Physical settlement of all contracts would result in an ex-post unbiased andcontext-free settlement mechanism. Inability to short sell makes the design problem non-trivial.An important issue considered in this paper, neglected by other authors, is participation. Anysettlement mechanism should take into account this issue when making predictions regarding thesettlement price and agents’ payoffs. The main result of the paper is that any settlement mechanismthat is context-free and from designer’s ex-ante stand point sets an unbiased price is a posted price
27
mechanism. This mechanism sets price equal to the expected value of the defaulted bond condi-tional on designer’s information. Moreover, this mechanism is almost surely the unique mechanismthat satisfies enumerated properties in the introduction. Moreover it guarantees participation by allagents.
The tool that is developed in this paper is a new approach to extend the notion of core to the caseof incomplete information. I considered the ’exit game’ before joining the blocking mechanism.This model can be applied to mechanism design problems in which (i) agents are allowed to gettogether and use another mechanism for their purpose and (ii) the decision to leave the mechanismis before formation of the grand mechanism. An example of such environments is dark markets inthe stock exchange.
28
9 Appendix
9.1 Proof of Inequalities in the Leading Example:One can rewrite agent i’s payoff when he receives when qi contracts are settled physically withprice pi as
Proof. Note that the second part of proposition 5.1 can be generalized to the case where the mech-anism is payoff equivalent to a posted price mechanism when the network is M. All one needs todo is to replace the mechanism with the posted price mechanism when the network is M and applyproposition 5.1.Taking expectation from the conclusion of proposition 5.1 implies:∑
i∈A
E[uNi (s)] ≥
∑i∈A
E[uMi (s)] for all s ∈ S . (21)
Since the mechanism is weakly-unbiased, the following holds:∑i∈A
Eρ[uNi (s)] =
∑i∈A
Eρ[uMi (s)]. (22)
Equality (22) and proposition 5.1 imply that almost surely for all s ∈ S K the following holds:∑i∈A
uNi (s) =
∑i∈A
uMi (s). (23)
Note that since there are a finite number of signals, almost surely means for almost all functionalforms and agents’ priors. �
I prove the following lemma.
Lemma 9.1. If the mechanism (p, q, u) is ex-post individually rational, full information unravel-proof, anonymous, ex-ante weakly budget balanced, and weakly-unbiased, then almost surely (i)uN
i (s) = 0 for all signal profiles s ∈ S K and agents i ∈ K who do not have any CDS contract innetwork N and (ii) the mechanism is ex-post budget balanced.
30
Proof. Consider the contract matrix in which no agent has a CDS contract, namely the 0 contractmatrix. Note that q0
i (s) = 0 for all i ∈ K and s ∈ K, this is because: (i) the mechanism isanonymous, q0
i (s) = q0j(s) for all i, j ∈ K and s ∈ S K , and (ii) the mechanism clears the number of
bonds,∑
i∈K q0i (s) = 0. In network N, consider a block by all agents who have CDS contracts with
all of their contracts. Lemma 6.3 implies for all s ∈ S K the following holds:∑i∈K(N)
uNi (s) = 0. (24)
Ex-post budget balancedness implies for all i ∈ K \ K(N) and s ∈ S K :
uNi (s) ≥ 0. (25)
Since the mechanism is ex-ante weakly budget balanced, the following inequality holds:
Eρ[∑i∈K
uNi (s)] ≤ 0. (26)
Inequalities (24), (25), and (26) implies that almost surely for all i ∈ K\K(N) and s ∈ S K , uNi (s) = 0
and also∑
i∈K(N) uNi (s) = 0. Therefore,
∑i∈K uN
i (s) = 0 for all i ∈ K. �
I establish the induction base. If |K(N)| = 2, I prove uNi has the form described in the theorem.
Assume N is a network in which agents i, j ∈ K have n CDS contracts in which i is a protectionseller and j is a protection buyer. Lemma 9.1 implies ∀s ∈ S K : uN
i (s) + uNj (s) = 0. Moreover,
since the mechanism is budget balanced, the following holds:
qNi (s)(pN
i (s)) + qNj (s)(pN
j (s)) = 0. (27)
Equality (27) together with market clearing, qNi (s) + qN
j (s) = 0, implies pNi (s) = pN
j (s) = 0. Let N′
be a contract network in which i and j have n CDS contracts where j is a protection seller and i isa protection buyer. Since the mechanism is anonymous, the following holds:
(qNi (s), pN
i (s)) = (qN′j (s), pN′
j (s)) and (qNj (s), pN
j (s)) = (qN′i (s), pN′
i (s)).
Let si, s′i ∈ S i and s−i ∈ S K\{i}. Ex-post incentive compatibility implies:
Equation (30) implies that inequalities (28) and (29) must hold with equality. I show for all si, s′i ∈S i, s j, s′j ∈ S j, and sK\{i, j} ∈ S K\{i, j} the following holds:
qNi (si, s j, sK\{i, j}) = qN
i (s′i , s′j, sK\{i, j}). (31)
Since all incentive inequalities hold with equality, I have:
qNi (si, s′j, sK\{i, j}) = qN
i (s′i , s′j, sK\{i, j}) and (32)
qNj (si, s′j, sK\{i, j}) = qN
j (si, s j, sK\{i, j}). (33)
Since qNi (s) + qN
j (s) = 0, (32) and (33) imply that
qNi (si, s j, sK\{i, j}) = qN
i (s′i , s′j, sK\{i, j}).
Therefore, qNi does not depend on si and s j. Since the mechanism is anonymous, it does not
depend on s j for all j ∈ K. Let κN,M ⊆ κ be the set of functional forms and agents priors thatsatisfy the lemma 6.3 and κ′ ⊆ κ be the set that satisfies lemma 9.1. I prove the theorem for the set(⋂
M≺NκM,N) ∩ κ′. 13 Let networks N and N(i) be the set of agents that has contracts with agent i,i.e., N(i) = { j ∈ K|ni, j , 0}. Let fN =
∑i∈K |N(i)|, I prove by induction on min{ fN , |K(N)|}. So far,
I have established the result for fN ≤ 2 and |K| ≤ 2. Therefore the result is true for all networks Nsuch that min{ fN , |K(N)|} ≤ 2. Given network N where min{ fN , |K(N)|} > 2 and i ∈ K(N), I showuN
i has the form described in the theorem. There are three cases:
1. N(i) = { j} for some j ∈ K and j has contracts with some other agents.Proof: I construct contract matrix N0 from N by removing all contracts except for contractsthat are between agents i and j (see Fig 4). Induction base for network N0 implies:
uN0
i (s) = qni, j(E[v|s] − Eρ[v]).
Lemma 6.3 for networks N and N0 implies uN0
j (s) =∑
l∈K\{i} uNl (s) for all s ∈ S K(N) (see
Figure 4). Corollary 5.2 implies∑
i∈K uNi (s) = 0 for all s ∈ S K , therefore uN
i (s) = uN0
i (s) =
qni, j(E[v|s] − Eρ[v]) for all s ∈ S K . This proves the result for this case.
13Note that the complement of this set is of measure zero.
32
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6
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ri r
r65 4
3
i1
i2
2
Network N
- rj
ir65
Network N0
Figure 4: this is an example of network N, N0 and the blocking in case1.
+ rj����
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Blocking mechanism
r
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3
i1
i2
2
2. N(i) = { j} for some j ∈ K and j has no contract with other agents.Proof: Without loss of generality, assume j is a protection seller. Since |K(N)| > 2, assumeagents i1, i2 ∈ K(N) \ {i, j} have a number of CDS contracts (see Fig 5). Starting from thecontract network N, I construct a network N′ as follows: remove the contracts between i andj, add ni, j contracts between j and i1 and ni, j contracts between i1 and i, where j and i1 areprotection sellers (see Fig 5). Since the mechanism is robust with respect to the network ofcontracts, uN
l (s) = uN′l (s) for all s ∈ S K and l ∈ K. I Apply step 1 for the network N′ and
complete the proof of this step.r
j
6ri r
r65
i1
i2
2
Network NFigure 5: this is an example for case 2.
rj����
Network N′
r
r65
i1�
i2
2
ir 5
3. Agent i has contracts with more than one agent, i.e., |N(i)| > 1.Proof: Let j, i1 ∈ N(i). I remove contracts that are between i and j and name the new networkN1. Network N2 is constructed from N by removing the contracts between i and i1. Removecontracts between i & j and i & i1 to construct network N3. Induction hypothesis impliesthat the theorem is true for N1, N2, and N3. Since j has the same contracts in N1 and N2, andi1 has the same contracts in N2 and N3, for all s ∈ K(N), I have:
uN2
j (s) = uN1
j (s), (34)
uN3
i1 (s) = uN2
i1 (s). (35)
33
Lemma 6.3 for N & N1, N & N2, and N & N3 implies for all s ∈ S K the following holds:
uNi (s) + uN
j (s) = uN1
i (s) + uN1
j (s), (36)
uNi (s) + uN
i1 (s) = uN2
i (s) + uN2
i1 (s), (37)
uNi (s) + uN
i1 (s) + uNj (s) = uN3
i (s) + uN3
i1 (s) + uN3
j (s), (38)
uN1
i (s) =∑
k∈K\{ j}
qni,k(E[v|s] − Eρ[v]), (39)
uN2
i (s) =∑
k∈K\{i1}
qni,k(E[v|s] − Eρ[v]), (40)
uN3
i (s) =∑
k∈K\{ j,i1}
qni,k(E[v|s] − Eρ[v]). (41)
Add (36) and (37) and subtract (38) and apply (39), (40), and (41) to imply:
uNi (s) =
∑k∈K
qni,k(E[v|s] − Eρ[v]).
This proves the result for this case.
9.3 Proof of Theorem 6.4I establish the result of lemma 6.3 for the case of incomplete information unravel-proof mecha-nisms. Let networks M,N satisfying M ≺ N, and assume pM
i and qMi do not depend on the reported
messages. If for some sA ∈ S A the inequality
E[∑i∈A
uNi (s)|πA(s) = sA] < E[
∑i∈A
uMi (s)|πA(s) = sA]
holds, then one can design a blocking mechanism similar to proposition 5.1 . This implies for alls ∈ S K the following holds:
E[∑i∈A
uNi (s)|πA(s) = πA(s)] ≥ E[
∑i∈A
uMi (s)|πA(s) = πA(s)].
Since the mechanism is unbiased, for all s ∈ S K
E[∑i∈A
uNi (s)|πA(s) = πA(s)] ≥ E[
∑i∈A
uMi (s)|πA(s) = πA(s)]. (42)
34
Assumption 6.1 implies almost surely for all s ∈ S K∑i∈A
uNi (s) =
∑i∈A
uMi (s).
Similar argument in the proof of lemma 6.3 implies the following: for all i ∈ K \ K(N) and allagents’ prior µ such that (E[v|s], µ) ∈ κ the equality Eµ[uN
i (s)] = 0 holds. Assumption 6.1 impliesuN
i (s) = 0 for all networks N and agents i ∈ K \ K(N). This proves the result of lemma 6.3 inthis case. Proof of lemma 9.1 is followed from lemma 6.3. The rest of the proof is identical to theproof of theorem 6.2.
9.4 Proof of Theorem 6.5• Step 1: I show the mechanism is almost surely ex-post budget balanced.
Proof: Consider the following participation choice: Given network N, for all s ∈ S K thatsatisfy
∑i∈K uN
i (s) < 0, agents choose the blocking mechanism designed as in proposition5.1. In this participation-choice, the designer observes the network N for all s ∈ S K thatsatisfy
∑i∈K uN
i (s) ≥ 0. Since the mechanism is weakly budget balanced, it must be thatfor all functional forms and almost all agents priors
∑i∈K uN
i (s) = or < 0. Now consider aparticipation choice in which all agents choose to participate when the network is N. Weaklybudget balancedness and the fact that there are a finite number of signal profiles, imply∑
i∈K uNi (s) = 0 for all agents i ∈ K and networks N.
• Step 2: Let networks M,N satisfy M ≺ N and A is the set of agents who reduced the networkN to M, if the mechanism is payoff equivalent to a posted price mechanism when the networkis M, then the following holds: ∑
i∈A
uNi (s) =
∑i∈A
uMi (s).
Proof: Let S be the set of type profiles for which the following inequality holds:∑i∈A
uNi (s) <
∑i∈A
uMi (s).
Consider a participation-choice in which agents in A block the mechanism and reduce thenetwork from N to M (see proposition 5.3). Since the mechanism is unbiased, for all i ∈ Kthe following holds:
Eρ[uMi (s)I{s∈S } + uN
i (s)I{s<S }] = Eρ[ni(100 − E[v|s])] (43)
35
Summing up theses equalities for all i ∈ K(N − M) and definition of S implies:
Eρ[∑i∈A
uMi (s)] ≥ Eρ[
∑i∈A
ni(100 − E[v|s])] (44)
= Eρ[∑i∈A
mi(100 − E[v|s])] (45)
Consider a participation-choice in which the M network does not unravel; unbiasednessimplies that (44) must hold with equality. Hence, for all s < S ,∑
i∈K(N−M)
uNi (s) =
∑i∈K(N−M)
uMi (s).
Assume S is non-empty for a positive measure of designer’s belief support. I have:
Eρ[∑
i∈K(N−M)
uNi (s)] < Eρ[
∑i∈K(N−M)
ni(100 − E[v|s])]. (46)
This contradicts with unbiasedness of the participation-choice in which the N network doesnot unravel. Therefore, almost surely for all s ∈ S K∑
i∈K(N−M)
uNi (s) =
∑i∈K(N−M)
uMi (s).
Using step 1 and 2 and replicating the proof of theorem 6.2, I conclude that the mechanismis almost surely of the form described in theorem 6.2.
9.5 Proof of Theorem 7.1Since the new notions of unbiasedness and unravel-proofness are stronger and step 2 of proof oftheorem 6.2 is the only part that uses the property of robustness with respect to the network ofcontracts, all I need to do is to replace this step. Here is the replacement:
1. N(i) = { j} for some j ∈ K and j has no contract with other agents and there is an agent whohas contracts with at least two agents.Proof: Construct N0 as in the proof of theorem 6.2. Lemma 6.3 for N and N0 impliesuN
i (s) + uNj (s) = 0 for all s ∈ S K . Note that the first induction base implies:
uN0
i (s) + uN0
j (s) = 0.
36
Let l ∈ K \ {i, j} be the agent who has contracts with two other agents. Assume l′ ∈ K \ {i, j}has contracts with agent l. Consider a coalition in which agent l′ sells all of his contracts thathe has with l to agent j and all other contracts except for those between i and j are settled inthe blocking mechanism. Let N1 be the reduced network. Note that K(N1) = {i, j, l}. I provethat the result is valid for N1. Given N1, remove contracts that i and j have. Induction baseand lemma 6.3 imply
uN1
i (s) + uN1
j (s) = qn1j,l(E[v|s] − Eρ[v]).
Similarly, uN1
j (s) + uN1
l (s) = qn1i, j(E[v|s] − Eρ[v]). Corollary 5.2 implies:
uN1
i (s) + uN1
j (s) + uN1
l (s) = 0.
Last three equalities imply the following equalities:
uN1
j (s) = qn1j,l(E[v|s] − Eρ[v]) + qn1
j,i(E[v|s] − Eρ[v]),
uN1
i (s) = −qn1j,i(E[v|s] − Eρ[v]) = qn1
i, j(E[v|s] − Eρ[v]),
and uN1
l (s) = −qn1j,l(E[v|s] − Eρ[v]) = qn1
l, j(E[v|s] − Eρ[v]]).
Therefore, the result is true for N1. I apply lemma 6.3 for N and N1 and conclude:∑k∈K\{i}
uNk (s) =
∑k∈K\{i}
uN1
i (s) for all s ∈ S K .
Proposition 4.3 implies for all s ∈ S K :
uNi (s) = uN1
i (s) = qn1i, j(E[v|s] − Eρ[v]).
This proves the result for this case.
2. N(i) = { j} for some j ∈ K and j has no contract with other agents and there is no agent whohas contracts with at least two agents.Proof: Since |K(N)| > 2, assume agents l, l′ ∈ K(N) \ {i, j} have a number of CDS contracts.Construct N1 as in the previous case and N2 similar to case N1 with the difference that agentl sells all of his contracts to agent j. Similar argument as in case two shows the result is validfor N2. Lemma 6.3 implies for all s ∈ S K(N):∑
k∈K\{i,l}
uNk (s) =
∑k∈K\{i,l}
uN1
i (s),∑k∈K\{ j,l}
uNk (s) =
∑k∈K\{ j,l}
uN2
i (s).
37
Since unravel-proofness implies budget balancedness,∑k∈{i,l}
uNk (s) =
∑k∈{i,l}
uN1
i (s), (47)∑k∈{ j,l}
uNk (s) =
∑k∈{ j,l}
uN2
i (s). (48)
Subtracting (47) from (48) and applying uNi (s) + uN
j (s) = 0 implies
uNi (s) =
∑k∈{i,l}
uN1
i (s) −∑
k∈{ j,l}
uN2
i (s).
Since agent l has the same number of contracts in N1 and N2,
uN1
l (s) = uN2
l (s).
Since the result is true for N1 and N2, the following holds:
uN1
i (s) = qni, j(E[v|s] − Eρ[v]), (49)
uN2
j (s) = qn j,i(E[v|s] − Eρ[v]). (50)
Hence, uN1
i (s) = qni, j(E[v|s] − Eρ[v]).
9.6 Proof of Proposition 7.2The unbiased and weakly-unbiased properties have been used in equations (22), (42), and (43).These equations still hold if I make the replacement. Also it has been argued that equation (46)contradicts with the unbiasedness property. This equation also contracts with ex-ante uniform priceproperty.
38
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