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6 1752 Management Science 58(9), pp , 212 INFORMS make the following simplifying assumption about the information structure. Assumption 4.2. For each n, the value of the portfolio to the nth broker is given by v n = v q n. The random variables 1 N are independent and identically distributed with bounded support, denoted by, and are independent of the random variable q, which we assume to also have bounded support. Both are positive with probability one and have nonzero measure. Furthermore, the seller only observes q and each nth broker only observes n. The parameter q captures characteristics of the portfolio driving its desirability to brokers, and n is a parameter that captures the sensitivity of broker n to q. In reality, of course, a given portfolio may have associated with it a vector of relevant descriptors q = q 1 q D, where q d R and D is the number of descriptors, and we might think of a given broker s valuation as a function mapping R D R. If we make the assumption that the broker s valuation function is of the form v q = D d=1 d q d, where d is a decreasing linear function for the dth descriptor, then we can equivalently represent q as a single descriptor q R that describes the extent that q lies along the direction of greatest decrease of v. The reduction of q R D to q R thus amounts to a coordinate transformation. It is this concept that we are trying to compactly represent by using a single aggregate portfolio descriptor q. Because n is the only payoffrelevant information that is unique to the nth broker, this parameter can be thought of as the broker s type. Let us offer a more concrete interpretation of the terms defining v n. Think of v as the spot value of the portfolio, which is the commonly known market value that the seller could receive if he could liquidate without incurring transaction costs. The value v n of the portfolio to a broker is generally less than this. We take v to be a fixed constant known to all participants, true for an earlier spot but an allowable simplifying assumption for the case of a future spot because any uncertainty in v would be present in either auction mechanism. This assumption is easily removed if desired in our model and can be thought of as reflecting the trend of sellers increasingly preferring past spots. The parameter q, representing portfolio characteristics that determine its value to brokers, can, for example, be thought of as representing the liquidity of assets in the portfolio. Note that this particular interpretation is for illustration and neither necessary nor all-inclusive. If q =, the assets are perfectly liquid and v n = v. The sensitivity parameter n in turn reflects how rapidly the broker s assessment of the portfolio s value decays with increasing illiquidity. It may depend, for example, on features of the broker s execution technology and trading expertise. Under this interpretation, the term n q hence represents the transaction costs incurred by the broker from the time the portfolio is acquired until it is completely liquidated, i.e., the difference in realized value when the broker trades out of the portfolio relative to its spot value v when received from the seller. It is useful to think about the value of the portfolio to broker n prior to observing q in terms of the certainty equivalent under Assumption 4.1, i.e., v ce n = 1 r ln( E e rv n n ) (1) Note that, given only knowledge of n, the nth broker is indifferent between receiving the portfolio and receiving a fixed sum of money v ce n. Furthermore, if the broker pays an amount b for the portfolio, the certainty equivalent value of the transaction is 1 r ln( E e r v n b n ) = v ce n b 5. Mechanisms In this section we present two auction formats for which we will derive pure strategy bidding functions for a Bayesian-Nash equilibrium, the assumed solution concept we use to predict broker behavior. We begin with the standard first-price auction and then describe its counterpart that makes use of a trusted intermediary Standard First-Price Auction In the first-price auction, each broker places a bid and the portfolio is sold to the highest broker at the highest bid price. Though this does not affect our analysis, it is worth noting that this may be thought of as a sealed bid auction as each broker generally would not want others to see his bid. As a model of broker behavior, we will assume brokers behave according to pure strategy bidding functions that form part of a Bayesian-Nash equilibrium. Specifically, given N, r, and v and the symmetric nature of the game, we will assume that brokers employ a symmetric strategy that takes the form of a function S that maps the private information n of each nth broker to a bid S n, S R R. The subscript here indicates association with the standard first-price auction. Note that in this section it is assumed that brokers know the value of N as discussed in 4. Although in general one might hypothesize the existence of multiple symmetric equilibria as well as possible asymmetric equilibria, we will argue invoking prior results in the literature that there exists a unique Bayesian-Nash equilibrium bidding function,

7 Management Science 58(9), pp , 212 INFORMS 1753 which is symmetric, for this auction. As such, this bidding function S uniquely satisfies [ ( ( )) S n arg max E u v n b 1 b max S m n b R m n for all n. We begin with the following preliminary lemma. Lemma 5.1. Let Assumptions 4.1 and 4.2 hold. There exists a unique pure strategy symmetric Bayesian-Nash equilibrium bidding function for the standard first-price auction. This bidding function is strictly decreasing on and is differentiable. Proof. This follows from Theorem 2 in Maskin and Riley (1984), the conditions for which our model may be shown to meet. We now derive an expression that characterizes this equilibrium bidding function and will facilitate our computational study. Theorem 5.2 (Brokers Bidding Function in a Standard First-Price Auction). Let Assumptions 4.1 and 4.2 hold and assume that the distributions of broker types n and the portfolio characteristic q have densities. In a standard first-price auction, the unique pure strategy symmetric Bayesian-Nash equilibrium bidding function is given by S n = v ce n + 1 r ln [ 1 re rv ce n vce n F ce v ce n N 1 F ce N 1 e r d (2) where F ce is the cumulative distribution function of each broker s certainty equivalent value. Proof. The symmetric equilibrium bidding function satisfies n arg max E [ e r v n b 1 b max m n m n b R arg max P b R ( ) b max m m n (E [ e r v n b n + 1 ) 1 for all n, where we temporarily drop the subscript S to simplify notation. By Lemma 5.1, there is a unique solution to these inclusions, and it is monotonically decreasing on and differentiable. Let denote the image of generated by. Note that there is an injective mapping between n and values of v n ce v ce n that allows us to reinterpret equivalently as a mapping from brokers certainty equivalent valuations to bids. Let Ṽ ce denote the image of under the transformation to certainty equivalent value. We now view with this interpretation and will first solve for the bidding function in terms of vce n for convenience, and then as a final step rephrase this in terms of n via substitution. An equivalent condition on the Bayesian-Nash equilibrium bidding function is thus given by ( ) ( e v n ce argmax P b max b R m n vm ce r vce n b +1 ) 1 Because the mapping from n to vce n is injective, Lemma 5.1 also applies to this reinterpretation of as a function of vce n, providing strictly increasing monotonicity in vce n and differentiability. Because is monotonically increasing in vce n, it has an inverse that we will denote by. Note that is differentiable on. A broker who bids b and assumes the other N 1 brokers employ the equilibrium bid function believes that he will win the auction with probability F CE b N 1, i.e., the probability that all other brokers certainty equivalent valuations lead to bids lower than his. We therefore have ( ) ( e P b max m n vm ce r vce n b + 1 ) 1 = F CE b N 1( e r vn ce b + 1 ) 1 If b = v n ce for some vn ce Ṽ ce, it must satisfy the first-order condition for Bayesian-Nash equilibrium: = d db { FCE b N 1[ e r vn ce b } = N 1 F CE b N 2 d db F ce b [ e r vn ce b F CE b N 1[ re r vn ce b = N 1 F CE b N 2 f ce b b [ e r vn ce b + 1 rf CE b N 1 e r vn ce b Substituting v n ce v ce, b v ce, b v ce, b 1/ v ce, and d/db F b f b b in the above first-order condition and rearranging terms, we arrive at N 1 F CE v ce N 2 f ce v ce [ e r v ce v ce + 1 r v ce F CE v ce N 1 e r v ce v ce = Let v ce = e r v ce F ce v ce N 1 so that = rf ce v ce N 1 v ce e r v ce + N 1 F ce v ce N 2 f ce v ce e r v ce Substituting this expression for v ce into the prior equation, we obtain N 1 F ce v ce N 2 f ce v ce = v ce e rv ce

8 1754 Management Science 58(9), pp , 212 INFORMS Solving for v ce, we have v ce = e rv ce N 1 F ce v ce N 2 f ce v ce = e rv d ce F dv ce v ce N 1 ce and noting that =, alternative expression for is then given by v ce = vce = e r df ce N 1 We can now derive the desired expression for the bidding function by substituting this into the expression defining v ce. This gives us v ce = 1 [ r ln v ce F ce v ce N 1 = 1 [ r ln 1 vce e r df F ce v ce N 1 ce N 1 = = 1 N r ln F ce v ce + 1 [ r ln vce e r df ce N 1 We now integrate by parts, letting dv = df ce N 1 v = F ce N 1 and u = e r du = re r d. Making the appropriate substitutions gives us v ce = 1 N r ln F ce v ce + 1 r ln [ e r F ce N 1 v ce = 1 N ln F r ce v ce + 1 { [e r ln rv ce F ce v ce N 1 = 1 N r ln F ce v ce +v ce + 1 r ln [F ce v ce N 1 = v ce + 1 r ln [ 1 re rv ce = vce vce vce vce F ce v ce N 1 = F ce N 1 e r r d } F ce N 1 e r vce r d F ce N 1 e r vce r d F ce N 1 e r d Finally, reintroducing the dependence on and the subscript as well as specifying a particular broker n, we have our result S n = v ce n + 1 [ r ln 1 re rv ce n F ce v ce n N 1 vce n F ce N 1 e r d Although the above discussion has focused on the derivation of the unique pure strategy symmetric Bayesian-Nash equilibrium bidding strategy, it turns out that this is also essentially the unique pure strategy Bayesian-Nash equilibrium. In particular, as the following result establishes, if there is an asymmetric pure strategy Bayesian-Nash equilibrium, the bidding function must be equal to that of the pure strategy symmetric Bayesian-Nash equilibrium except possibly for a set of types of measure zero. Theorem 5.3. Let Assumptions 4.1 and 4.2 hold and assume that the distribution of broker types n has a density. In a standard first-price auction, there exists a unique pure strategy Bayesian-Nash equilibrium bidding function up to a set of types of measure zero. Proof. It may be shown that our model satisfies the assumptions of Theorem 1 in McAdams (27), which establishes existence and uniqueness of a pure strategy Bayesian-Nash equilibrium. Uniqueness of the equilibrium bidding function increases confidence in our derived Bayesian-Nash equilibrium as a model of broker behavior Intermediated First-Price Auction In the intermediated first-price auction, a trusted party solicits from the seller a description of the portfolio and from each broker a function that specifies his bid contingent on the portfolio. In the context of our model, the portfolio s description can be thought of as the parameter q and each broker s bid function n I is a function of q and the number of brokers N. Note that unlike in the standard auction, here N is known to the intermediary and may be reflected in bid values without the requirement that its value be known to brokers, which we do not assume. The intermediary assigns the portfolio to the broker who offers the highest bid n I q N and the portfolio is purchased at this price. The nth broker s choice of bid function n I should depend on his type n. Alternatively, we can view the bid function as a function of three variables, with n I n q N being the bid the broker assigns given his value of n, the portfolio parameter q, and the number of brokers N. With this perspective, what the broker supplies to the intermediary is a function n I n of the portfolio parameter and broker number. We will view the strategy employed by each broker as such a function of three variables. As a solution concept, we assume as before that brokers behave according to a pure strategy Bayesian-Nash equilibrium. It is easy to extend results from the previous section to show that there exists a unique pure strategy Bayesian-Nash equilibrium, which is symmetric.

9 Management Science 58(9), pp , 212 INFORMS 1755 We therefore drop the superscript n. In particular, for each n, I n q N arg max E b R [ ( u v n b ( )) 1 b max I m q N n q N m n The following result serves a purpose analogous to Theorem 5.1 but in the context of an intermediated auction. Theorem 5.4 (Bayes-Nash Equilibrium Bids in an Intermediated First-Price Auction). Let Assumptions 4.1 and 4.2 hold and assume that the distribution of broker types n has a density. In an intermediated first-price auction, the unique pure strategy Bayesian-Nash equilibrium bidding function up to a set of types of measure zero is given by I n q N = v n + 1 [ r ln 1 re rv n F v v n N 1 v n F v N 1 e r d (3) where v n = v n q is a function mapping broker types to broker valuations and F v is the cumulative distribution of these valuations. Proof. The proof is analogous to that of Theorem 5.1 but now making use of the full information available to the intermediary when calculating brokers bids. Uniqueness again follows from Maskin and Riley (1984) and McAdams (27). 6. Benefits of an Intermediary Based on the derived Bayesian-Nash equilibria bidding strategies for the standard and intermediated first-price auctions derived in 5, several numerical studies were performed to assess how these two mechanisms compare regarding the seller s transaction costs. This was done via Monte Carlo averaging over specific values for q and n drawn from their respective distributions. In each case, the constants characterizing a given study are N and r. In the following subsections we first discuss the motivations behind our choices for the specific parameter values used in the numerical studies and then present our results Representative Instances We will discuss results of computations carried out to assess the benefits afforded through use of an intermediary. These computations involve model instances with specific parameter settings. In particular, we make Assumptions 4.1 and 4.2 and consider instances in which v = 5 1 8, q unif 1 7, and n unif 1. We consider ranges for the number of brokers N 2 12 and the risk-aversion parameter r Let us motivate these choices. The values of v and the ranges for q and n are chosen to reflect a realistic portfolio value of half a billion dollars and a realistic range from zero to ten million dollars of potential broker transaction costs. Blind portfolio auctions typically engage two to four dominant brokers. The range we consider for N includes these possibilities and extends beyond them. The choice of risk-aversion parameter deserves further discussion. The case of risk neutrality (r = ) serves as one extreme worth understanding for insight though it is unrealistic. Our intention is for realistic risk-aversion parameter values to be captured within the range To understand why this intention should be served, consider a situation where a broker is uncertain about what a portfolio is worth to him and assumes a normal distribution with a standard deviation of one million dollars around his expectation. What premium would have to be subtracted from his expectation to arrive at a price where he is indifferent about acquiring the portfolio? Different brokers would have different opinions on this matter, but a representative figure might be one hundred thousand dollars, which is 1% of the standard deviation. This guess implies a risk aversion of r = In particular, letting the broker s profit, which is the difference between the amount he pays for the portfolio and the amount he later discovers it is worth to him, be denoted by x N , this value of r solves v CE n = 1 r ln( E e rx ) = 1 5 r = 6.2. Benefits to the Seller Prior to participating in the auction, the seller holds a portfolio with a spot value of v, i.e., the portfolio value at the time at which the trade price is being fixed, such as the market closing price, etc. Note that this nominal value is assumed known by each of the N brokers, i.e., common market knowledge. Through participation in the auction, the seller is able to liquidate this portfolio by selling it to the winning broker, whose bid we will denote by b W. When liquidating the portfolio, the seller s goal is to do so while minimizing the transaction costs incurred in the process. Letting T denote the transaction costs of the seller, we have that T = v b W, representing the difference between the book value of the portfolio and what the seller receives for it. As our model assumes a risk-neutral seller, the metric of concern from the seller s point of view is the expected transaction cost E T. Computing the brokers Nash equilibrium bidding strategies using

10 1756 Management Science 58(9), pp , 212 INFORMS the expressions derived in 5, we are able to numerically compute E T and study how the standard and intermediated first-price auctions compare as we vary two parameters, the brokers common risk-aversion parameter r, as well as the number of brokers N. The importance of studying the change with respect to r is immediate in that this gauges how relevant the informational asymmetry between the seller and brokers is as seen from the brokers point of view and how this in turn affects the seller. The number of brokers N from which the seller receives bids is important in that it represents the only degree of freedom, or operating parameter, left to the discretion of the seller once he has chosen either the standard or intermediated mechanism. As mentioned in 1, typically sellers collect bids from a small handful of brokers, often on the order of 2 4, to avoid information leakage. By introducing a trusted intermediary, sellers are safely able to solicit bids from an arbitrarily large number of brokers without risk. Given this possibility, it is important to consider the potential benefits of higher values of N made possible by an intermediary and compare this to the standard case. Consider Figures 2 and 3, which compare the average seller transaction cost in standard and intermediated first-price auctions as r and N are varied, Figure 2 Average Seller Transaction Cost vs. Risk Aversion (95% Confidence Intervals) N = r 1 7 r N = 8 Standard, first price Intermediated, first price respectively. There are several points that summarize our findings. In what follows, we will use T A to denote the average seller transaction cost when the auction is of type A S I, denoting a standard or intermediated auction. We see that across all risk-aversion levels r > that T I < T S and that the gap between the two increases with r. There are in essence two pressures driving broker behavior as risk aversion grows. On the one hand, increased risk aversion leads to the possibility for additional expected utility gains from bidding higher, resulting from marginal probability of winning gains that outweigh marginal utility losses when the bid is increased. This tends to push the expected seller revenue up and hence improves the seller s transaction cost. However, as r increases, v ce for each broker decreases as well, resulting in lower bids, lower expected seller revenues, and hence higher transaction costs. Which force dominates will depend on the degree of uncertainty in the portfolio being auctioned and this will determine how increasing r affects the seller s transaction costs. In the case of the standard auction, the portfolio uncertainty dominates and hence increasing r increases T S. Interestingly, the other pressure dominates in the intermediated case and we see that T I decreases r 1 7 r N = 4 N = 12

11 Management Science 58(9), pp , 212 INFORMS 1757 Figure 3 Average Seller Transaction Cost vs. Number of Brokers (95% Confidence Intervals) r = 1e N r = 2e 7 Standard, firs price Intermediated, first price N as r increases. To explore this, consider an intermediated auction under Assumption 4.2. We will refer to one utility function u 2 as being more risk averse than a second u 1 if it can be represented as a concave transformation of the second, i.e., u 2 = u 1, where is an increasing concave function. Let r A x u denote the Arrow-Pratt coefficient for absolute risk aversion of a utility function u at x and let c F u denote the certainty equivalent of a gamble F under utility function u. Then it may be shown (Mas-Colell et al. 1995) that this relationship between utility functions is equivalent to each of the individual statements that r A x u 2 r A x u 1 x and c F u 2 c F u 1 for any F. In addition, we define U to be the set of differentiable monotonically increasing utility functions such that u =. The following theorem establishes in a fairly general setting that seller transaction costs decrease as broker risk aversion increases, as observed in our numerical study. Although we focus on utility functions in U in this result, note that it applies to any family of utility functions that may be transformed into U via a constant offset as this is irrelevant to the study of relative mechanism N r = 1e 7 r = 4e N dynamics. This thus applies to the utility form adopted in Assumption 4.1. Theorem 6.1 (Transaction Costs Decrease as Risk Aversion Increases in an Intermediated Auction). Let Assumption 4.2 hold. For all u 1 u 2 U such that u 2 is more risk averse than u 1, T I is strictly less if brokers realize utility u 2 than if brokers realize utility u 1. Proof. Note that an intermediated auction is conceptually equivalent to a first-price auction where q is common knowledge among all brokers and each broker bids optimally given that information. We proceed under this framework. Assume that in equilibrium there exists a monotonically increasing and differentiable bidding function mapping values to bids with the property that =. Focusing on a given broker with value v, if all other brokers follow the equilibrium strategy then the broker solves for their optimal bid value according to max H z u v z z where z functions as the broker s decision variable when determining his bid and H is the distribution of the highest value of the other N 1 players,

12 1758 Management Science 58(9), pp , 212 INFORMS also assumed differentiable. Taking the first derivative with respect to z gives us h z u v z H z u v z z = or in equilibrium when brokers bid according to their true values, v = h v u v v H v u v v (4) Now consider two different risk-averse utility functions u 1 u 2 U such that u 2 represents a greater degree of risk aversion, i.e., we may relate the two via u 2 = u 1, where is an increasing concave function. Using (4), we have h v u 2 v = 2 v 2 v H v u 2 v 2 v = h v u 1 v 2 v H v u 1 v 2 v = h v u 1 v 2 v H v u 1 v 2 v u 1 v 2 v (5) From our assumption that u 1 u 2 U, implying u 1 = u 2 =, it follows necessarily that =. Using this fact along with the concavity of, it may Figure 4 Two Utility Functions, the More Risk-Averse One Normalized in the Bottom Plot Such That u 2 = and u 1 v v = u 2 v v Utility Utility Payoff be easily shown via a first-order Taylor expansion that x / x > x for all x in its domain. Applying this inequality to (5) gives us h v u 2 v = 1 v 2 v H v u 1 v 2 v u 1 v 2 v > h v u 1 v 2 v H v u 1 v 2 v (6) Now note that (4) may be also applied to u 1 to give us h v u 1 v = 1 v 1 v H v u 1 v 1 v (7) Consider now the scenario in which 1 v > 2 v. This leads to the relationship that v 2 v > v 1 v. Because u 1 is defined to be concave, u 1 x and u 1 x as x. Hence, we may conclude that for 1 v > 2 v that h v u 2 v > 1 v 2 v H v u 1 v 2 v > h v u 1 v 1 v H v u 1 v 1 v = 1 v Thus, we have shown that 1 v > 2 v 2 v > 1 v. Combining this with the knowledge that 1 = 2 =, we have that 2 v > 1 v v >. Because Less risk averse More risk averse.2 Less risk averse More risk averse (normalized, equivalent) Payoff Note. Here v v = 1 corresponds to the optimal bid v for the less risk-averse utility function.

13 Management Science 58(9), pp , 212 INFORMS 1759 for any given value v, brokers with strictly greater risk aversion will compute strictly larger equilibrium bids, it follows that the corresponding seller revenue will be strictly greater. Finally, because the seller s transaction costs decrease with increasing revenue, it follows that T I strictly decreases. The intuition behind Theorem 6.1 may be seen in Figure 4. In the top plot we see two utility functions, one more risk averse (dashed curve) than the other (solid curve). Without any change in broker behavior, we may shift and scale the dashed curve, as seen in the bottom plot, to start from the origin and cross the solid curve at the point corresponding to the solid curve s utility when a broker in that scenario wins the auction by bidding the optimal amount v when their valuation is v. When normalized this way, the slope of the less risk-averse utility function is always greater than that of the more risk-averse utility function at that point. As a result, the optimal bid for the more risk-averse case must be greater than v as the marginal decrease in expected utility as the bid increases is less, i.e., the optimal point where the marginal decrease in expected utility balances the marginal gain in winning probability will occur for a larger optimal bid. Interpreting these results in terms of representative parameters discussed in 6.1, r = and the inclusion of an intermediary provides the seller with a transaction cost savings of over 1% relative to the standard auction for the case of N = 2. This represents a significant improvement for institutional investors. With respect to changes in the number of brokers N, it is observed that for both the standard and intermediated cases and for all r > that the seller s transaction costs are convex in N and asymptotically approach a common fixed lower bound as N increases. This result is driven by two factors. First, as N increases, min n n, and hence in both the standard and intermediated cases the value of the winning broker approaches v. Second, as N increases, the degree to which a broker in a first-price auction will bid below his valuation decreases in order to remain competitive. In the limit, brokers in both auctions bid asymptotically closer to v as N increases. It is the ability of the intermediated case to prevent information leakage that allows sellers to increase N to approach this limit arbitrarily closely, minimizing their transaction costs. 7. Intermediary Implementation An important issue is how to implement the notion of a trusted intermediary. For the auction mechanism to function properly, participants need to be guaranteed that the information revealed to the intermediary (i.e., bidding functions I n from brokers and portfolio characteristic q from the seller) will be kept completely confidential and used appropriately. If that is not the case, then the proposed system and its merits will decay because of two factors. First, brokers will become increasingly hesitant to participate in the market if they feel that their information may be compromised, denying both the seller and brokers the opportunity to potentially gain from a principal trade. The second result of a lack of trust in the intermediary is that risk-averse brokers will factor the potential for information leakage into their bids, which will be subsequently worse for the seller. One possible form for an intermediary would be a government sanctioned organization, perhaps itself a division of the U.S. Securities and Exchange Commission, that has sufficient transparency and auditory protocols in place to ensure information security. This could also be a role potentially made available to nongovernmental third party companies, and competition among them could lead to the evolution of a protocol for handling these transactions that best satisfies participants needs. The ability for participants to verify the results of auctions when necessary would be important, and various details regarding issues like participant anonymity, etc. would need to be carefully considered. Note that the finance industry has already seen the advent of third party companies that facilitate portfolio trades, though these companies generally get involved in sizable transition trades where a seller is making a large shift in assets or perhaps a pension fund has changed management. A second manner in which the intermediary may be implemented is via an equivalent cryptographic protocol. In the past decade, researchers in computer science have begun looking at ways in which concepts from cryptography, namely, zero-knowledge proofs and related protocols, may be applied to various settings in electronic commerce, auctions, and security exchanges. In Thorpe and Parkes (29), for instance, a methodology is presented by which a seller s order may be completely filled, making use of a secure and verifiably accurate way to report to potential brokers the risk parameters of a portfolio formed by combining their current positions and those of the seller, with neither party knowing the positions of the other. In addition, Izmalkov et al. (28), Micali (21), and Izmalkov et al. (211) present theoretical work that develops methodologies by which mechanisms and trusted intermediation may be implemented by the agents themselves in a manner that is verifiable and without anyone s private information being divulged. Although currently limited in practicality when applied to arbitrarily complicated mechanisms, this line of research suggests that further work will allow for a distributed cryptographic implementation of an intermediary in the future.

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