# Efficiency and Information Transmission in Bilateral Trading

Abstract: We study pairwise trading mechanisms in the presence of one-sided or two-sided private information and two-sided limited commitment, whereby either trader can walk away from a proposed trade when he learns the trading price. We show that when one trader's information is relevant for the other trader's value of the asset, optimal trading arrangements may necessarily conceal the traders' information. While limited commitment itself may not be costly, it shapes how prices transmit information.

## Summary (4 min read)

### 1 Introduction

- If existing arrangements strive for trading efficiency, then their results imply that information transmission may be purposefully prevented in the marketplace.
- A key attribute of these markets is that many buyers and sellers are financially sophisticated traders who, under certain circumstances, may have private information about an asset’s cash flow.
- A distinguishing feature of their approach is that the authors also impose ex post participation constraints: after the mechanism recommends that trade takes place at a particular price, both traders have the option to walk away, given all the information available at that time.
- Even in cases where the no-commitment and commitment allocation coincide, the implications for information revelation are different.

### 2 One-Sided Private Information

- One of the traders, the seller, initially owns the asset.
- Throughout this section, the authors assume that the seller privately observes a signal and so has more information than the buyer.
- The one-sided private information assumption is perhaps harder to justify in the context of financial markets, since buyers and sellers can each observe some public information about the asset and can conduct research that may grant them access to other signals.
- Following Kennan and Wilson (1993), the authors view the mechanism design approach as informative about what traders may accomplish through any mechanism, including bargaining, given the restrictions implied by private information and technology.

### 2.1 Interim Optimal Mechanisms

- The authors start by analyzing Pareto optimal trading mechanisms.
- Using the revelation principle (Myerson, 1979), the authors know that any trading mechanism is payoffequivalent to a direct revelation mechanism.
- The first constraint imposes that the buyer’s expected profit is at least u; by varying u the authors trace out the Pareto frontier between the expected profit for the seller and buyer.
- This implies that if a seller with some other signal s′′ has an incentive to truthfully report s′′ rather than s, he also has an incentive to truthfully report s′′ rather than s′.

### 2.2 Information Revelation and Ex Post Participation

- The mechanism design problem (1) is a technical device for characterizing possible trading arrangement given the constraints implied by private information.
- In their view, problem (1) ignores an important constraint on real world trading arrangements.
- The ex post participation constraint imposes that the buyer earns nonnegative profits at any trading price p, an additional constraint on feasible trading mechanisms.
- The authors view condition (2) as being quite weak.
- With such a mechanism, the buyer is left with a coarser information set than the seller, sometimes knowing only that the seller’s report was smaller than s1, other times knowing only that it lay in between s′1 and s2.

### 2.3 Cost of Full Information Revelation

- Proposition 2 establishes that, under certain circumstances, it is possible to implement the solution to problem (1) while satisfying the buyer’s ex post participation constraint.
- The buyer must also be willing to trade at that price, correctly interpreting how that price was influenced by the seller’s signal.
- The authors prove that full information revelation reduces the value of the seller.
- To show this, the authors modify problem (1) by adding one more constraint, q(s)b(s)− t(s) ≥ 0 for all s. (3) The buyer must earn nonnegative profit at each value of the seller’s signal.
- This approach is more convenient but has little practical impact on their results.

### 3 Two-Sided Private Information

- The authors turn next to the realistic case where both the buyer and seller have private information.
- The basic structure of the model is unchanged, except for the fact that the buyer now also has a signal.
- After briefly describing the economic environment, the authors formally discuss the constraints on mechanisms that arise from private information and the lack of commitment from both parties.
- The authors end the section by discussing the additional constraints imposed by full information revelation.

### 3.1 Preferences and Information

- Both the buyer and seller receive a signal and the authors assume for analytical convenience that the signal is binary, b ∈ {0, 1} denotes the buyer’s signal and s ∈ {0, 1} denotes the seller’s.
- Let πbs denote the ex ante joint probability that the buyer receives signal b and the seller receives signal s.
- The buyer’s expected value for the asset is vBbs when the buyer’s signal is b and the seller’s signal is s.
- But more generally, each trader’s willingness to trade depends on their belief about the other traders’ signals.
- Different trading mechanisms allow a trader to refuse to trade based on different information sets.

### 3.2 Feasible Mechanisms

- The authors are interested in understanding the set of feasible trades given the constraints imposed both by private information and by the ability of either trader to walk away from the deal after learning the terms of trade.
- This again motivates a mechanism design approach.
- The authors contrast this with the standard interim constraints, EB(vBbs − p|b) ≥ 0 and ES(vBbs − p|s) ≥ 0, which require that the trader’s expected profits are nonnegative only at the interim stage, before they learn the price recommended by the mechanism.
- The outer expectation is taken over prices, conditioning only on the trader’s own signal and report.
- As explained above, this is a constrained version of the standard mechanism design problem, which imposes only an interim participation constraint and an interim incentive compatibility constraint.

### 3.3 Formal Characterization of Mechanisms

- This section provides a formal characterization of veto incentive compatible mechanisms.
- There exists a feasible veto incentive compatible mechanism with the same trading probabilities and expected payoffs conditional on any signals (b, s), in which the recommendation is always of the form p ∈ {p1, p2, . . . , p13,∅}.
- 6Formally, the buyer’s ex post participation constraint follows from the buyer’s veto incentive compatibility constraints with b = b̂, and similarly for the seller.
- The finding that the authors use at most 13 prices depends on the dimension of the signals that the buyer and seller receive.
- The exact expression uses the observation that ωn|bsπbs is the probability that the price is pn, the buyer’s signal is b, and the seller’s signal is s.

### 3.4 Fully-Revealing Mechanisms

- A fully-revealing veto incentive compatible mechanism is a feasible veto incentive compatible mechanism in which the mechanism is constrained to reveal each trader’s report to the other trader before a trade is consummated.
- If the mechanism recommends trading at a price pbs when the buyer’s report is b and the seller’s report is s, then both the buyer and seller must be willing to trade at that price knowing the other trader’s signal.
- It is straightforward to verify that a fully-revealing mechanism needs to use at most one price per report pair (b, s), i.e. at most four prices in their scenario.
- The objective function is essentially unchanged from the interim and veto-incentive compatible problems.
- The final two constraints ensure that neither trader wishes to misrepresent his signal, allowing for the possibility that the trader then refuses to trade following certain reports by the other trade.

### 4 Private Values

- The authors briefly comment on the private values case, a common simplifying assumption in the mechanism design literature (Myerson and Satterthwaite, 1983; McAfee and Reny, 1992).
- If the buyer knows the seller has a low signal, he will be unwilling to pay a high price for the asset.
- In contrast, it may be possible to sustain more efficient trading arrangements by keeping some information private.
- The remainder of the paper shows how this works.

### 5 Common Values

- The authors now turn to the case with common values.
- The authors start by introducing a useful example which parametrizes the information structure in a way that lends itself easily to interpretation and assumes a constant gain from trade.
- The authors then solve for mechanisms that maximize the sum of utilities, or equivalently, the gains from trade.
- The authors then turn to mechanisms that impose full information revelation and show that this restriction is costly.
- Finally, the authors study other efficient mechanisms that trace out the Pareto frontier between the buyer and seller.

### 5.1 An Example

- The authors focus on a particular example which illustrates some general properties of the model.
- As motivation for the payoff structure, suppose that the state of the world is δ ∈ {0, 1}, taking on each value with equal probability.
- Here γ denotes the constant gains from trade.
- The signals are independent conditional on the asset’s payoff, but they are only imperfectly correlated with the payoff.
- To put a more economic interpretation on the example, the authors view the buyer’s and seller’s signals as models of the asset’s cash flow.

### 5.2 Trade Maximization

- The authors start by describing the solution to the veto incentive compatible problem (7) with equal Pareto weights, φ = 1/2.
- The following proposition characterizes the optimal mechanism: Proposition 7.
- Figure 1 shows the three regions of the parameter space.
- A seller with the high signal would refuse to sell at the intermediate price if she doesn’t know the buyer’s signal, and similarly a buyer with the low signal would refuse to buy at the intermediate price if he doesn’t know the seller’s signal.
- Any veto incentive compatible trading arrangement must reveal some information.

### 5.3 Fully Revealing Mechanisms

- The authors highlight the cost of information revelation by solving the full revelation problem (9).
- Conversely, in the third case in Proposition 7, the authors proposed implementing the optimum using a fully revealing mechanism.
- This is illustrated on the right hand side of the equality.
- As a result, the trading probability λ is lower when the buyer has the low signal and the seller has the high signal.

### 5.4 Pareto Frontier

- The authors close by exploring the set of feasible payoffs more generally.
- The basic issue is that an optimal mechanism in the interim problem may set a higher expected price when the buyer reports the low signal than when he reports the high signal: t0s/q0s > t1s/q1s for some s.
- The intermediate shaded region is the set of payoffs that can additionally be obtained using a veto incentive compatible mechanism, i.e. a feasible policy in problem (7).
- This gives the seller expected profit 0.214.

### 6 Conclusion

- This paper highlights the transmission of information within a single trade.
- In other cases, the unconstrained optimum is unattainable, but efficient trading arrangements still conceal information by making the trading price insensitive to information and possibly random conditional on all available information.
- Thus their basic approach trivially carries over to this environment.
- Minimizing the information that the insurance company learns from the mortgage originator enhances both trading and retrading efficiency.

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### Cites background or methods from "Efficiency and Information Transmis..."

...This condition is equivalent to λ ≥ λ∗, where λ∗ is given by 9 Recently, Shimer and Werning (2019) relax this commitment assumption....

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...6 For the mechanism that I use in the bilateral trading, I follow the literature on bargaining under asymmetric information using the mechanism design approach: Myerson (1979), Myerson and Satterthwaite (1983), Samuelson (1984), and recently Shimer and Werning (2019)....

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##### References

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### "Efficiency and Information Transmis..." refers background in this paper

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...A reduction in asymmetric information alleviates the lemons problem in Akerlof (1970), so one might expect that Pareto optimal trading mechanisms would induce traders to reveal their information to each other....

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...In response, Grossman and Stiglitz (1980) and Kyle (1985) introduced noise traders, alongside informed and uninformed traders....

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