# Fixed-Price Approximations in Bilateral Trade

We consider the bilateral trade problem, in which two agents trade a single indivisible item. It is known that the only dominant-strategy truthful mechanism is the fixed-price mechanism: given commonly known distributions of the buyer's value B and the seller's value S, a price p is offered to both agents and trade occurs if S ≤ p < B. The objective is to maximize either expected welfare 𝔼[S + (B-S) 1_S ≤ p < B] or expected gains from trade 𝔼[(B-S) 1_S ≤ p < B]. We determine the optimal approximation ratio for several variants of the problem. When the agents' distributions are identical, we show that the optimal approximation ratio for welfare is 2+√(2)/4. The optimal approximation for gains from trade in this case was known to be 1/2; we show that this can be achieved even with just 1 sample from the common distribution. We also show that a 3/4-approximation to welfare can be achieved with 1 sample from the common distribution. When agents' distributions are not required to be identical, we show that a previously best-known (1-1/e)-approximation can be strictly improved, but 1-1/e is optimal if only the seller's distribution is known.

## Authors

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## 1 Introduction

In this paper, we study the performance of fixed-price mechanisms in the canonical context of bilateral trade, in which a buyer and seller bargain over a single indivisible good. As is standard in the literature, agents have independent private values for the good, for the buyer and for the seller, and the mechanism designer is given some information about the distributions from which these values are drawn. We consider two basic settings in this paper: one where the distributions of the agents’ values are identical (the “symmetric” case), and one where the distributions are arbitrary (the “general” or “asymmetric” case). We measure the performance of a mechanism by considering either (i) the “gains from trade”: whenever trade occurs and otherwise, or (ii) the “welfare”: if trade occurs and otherwise. Observe that the difference between welfare and gains from trade is simply the seller’s value.

The “first-best” optimum is considered to be for welfare, and for gains from trade. In their seminal work, Myerson and Satterthwaite [Myerson] showed that no budget-balanced, individually rational Bayesian incentive-compatible (BIC) mechanism attains the first-best optimum in general. They also presented a “second-best” mechanism which is budget-balanced, individually rational, BIC, and achieves the best possible gains from trade attainable by any such mechanism. However, the second-best mechanism is complicated and difficult to implement in practical settings, so a recent growing literature seeks simpler mechanisms that maintain provably near-optimal approximation guarantees.

Motivated by the need for strategic simplicity, we focus on dominant-strategy incentive compatible (DSIC) mechanisms. A fundamental result of Hagerty and Rogerson [hagerty] is that essentially every DSIC mechanism for bilateral trade has the following form: the mechanism designer chooses a (possibly random) price and offers trade at this price to the buyer and seller. Trade occurs if and only if the buyer’s value exceeds the chosen price and the seller’s value does not; this mechanism is DSIC. We refer to this as a fixed-price or posted-price mechanism. Given this explicit characterization of the allowed mechanisms under our incentive compatibility criterion, other questions arise. How close to the efficient outcome can fixed-price mechanisms get? What information about the agents is required to set the optimal price? In this paper we give a comprehensive answer to these questions in the special case of symmetric bilateral trade, and partial answers in the general case.

### 1.1 Prior Work

Since the seminal work of Myerson and Satterthwaite [Myerson], a large literature has emerged that characterizes the optimal incentive-compatible mechanisms in various settings. When the strong budget balance condition is imposed, [hagerty] shows that dominant-strategy mechanisms for bilateral trade must essentially be fixed-price mechanisms, which we study in this paper for their strategic simplicity. [drexlkleiner15] and [shaozhou16] show that fixed-price mechanisms can be optimal even when the budget balance condition is relaxed to a no-deficit condition.

The two papers most closely related to ours are [Dobzinski-Blumrosen] and [efficientMarketsWithLimitedInformation]. The former shows that there is a randomized choice of a price based on the seller’s distribution which provides a -approximation to the optimal welfare (i.e., the expected welfare provided by the mechanism is at least times the optimal welfare). This approximation improves the bounds shown in previous work by [blumrosendobzinski14] and [colini-baldeschietal16]. On the other hand, [efficientMarketsWithLimitedInformation] shows that one can get a -approximation of welfare by just posting a sample from the seller distribution as a price (also in the asymmetric setting). Approximations to gains from trade have also been extensively studied in the literature [best-of-both-worlds, approximating-gft-in-bilateral-trading, strongly-budget-balanced, McAfee, FixedPriceApproximability, Kang]. In particular, [McAfee] shows that in the case of identical buyer’s and seller’s distributions, setting the price to be the median of the distribution achieves a -approximation to optimal gains from trade. More recently, [Kang] shows that choosing the mean also achieves a -approximation to optimal gains from trade, and this choice is in fact best possible among all fixed prices for a given distribution.

There have been various other results that show how first-best efficiency can be approximated when agents’ values are drawn from restricted families of distributions (i.e., distributions satisfying certain regularity conditions), such as those due to [arnostietal16] and [approximating-gft-in-bilateral-trading]. In this spirit, [shaozhou16] demonstrates the optimality of the mean as a fixed-price mechanism in a bilateral trade setting for log-concave distributions that have an increasing hazard rate.111Another difference between the result in [shaozhou16] and our Proposition 5 is that they do not restrict attention to posted-price mechanisms ex ante as we do, but rather consider dominant-strategy mechanisms with a no-deficit condition (as opposed to strong budget balance). By contrast, our objective is to consider the most general class of distributions possible. As such, our results do not depend on distributional assumptions; all we require is that the distribution of agents’ values has finite and nonzero mean.

### 1.2 Our Results

We extend the state of the art for bilateral trade in the following ways:

1. We show that a -approximation of optimal gains from trade in the symmetric case (which was already known by using the mean or median of the distribution) can be also achieved with the knowledge of a single prior sample from the distribution; by posting a price equal to this sample. Interestingly, our mechanism always achieves exactly of optimal gains from trade, for any distribution.

2. We prove that the same mechanism (posting a prior sample as a price) attains a -approximation to the optimal welfare in the symmetric case (compared to a -approximation by using a single sample in the asymmetric setting [efficientMarketsWithLimitedInformation]). is also tight for this mechanism.

3. We show that in the symmetric setting, a fixed-price mechanism using the mean as a price attains a -approximation of the optimal welfare, which is the best possible among fixed-price mechanisms.

4. For general (asymmetric) bilateral trade, with arbitrary agents’ distributions, we prove that there is a -approximation to optimal welfare for some constant , improving the previously best-known -approximation [Dobzinski-Blumrosen].

5. We also show that the -approximation cannot be improved with the knowledge of the seller’s distribution only (which is what the mechanism of [Dobzinski-Blumrosen] uses).

## 2 Model

Here we formally outline the model that was discussed above.

The bilateral trade problem is simple: two agents, a buyer and a seller, value an item that is held by the seller at and respectively. The specific realizations of and are unknown to us, but we assume that we have and , with and independent, for distribution functions and , about which we are given some information. For most of this paper, we assume that we have

A bilateral trade mechanism consists of an allocation function , which takes in as input the reported valuations of the buyer and seller and outputs 1 if a transaction should occur or 0 otherwise, and a payment function which, if a trade occurs, determines the price at which the item should be transacted. By the revelation principle [revelationPrinciple], we are interested in incentive-compatible mechanisms , where the agents are incentivized to report their true values. There are two standard notions of incentive compatibility that are considered in the literature: Bayesian incentive compatibility (BIC), and dominant-strategy incentive compatibility (DSIC). The former roughly says that reporting true values should be an optimal strategy for each agent, in expectation over the possible behavior of the other agent. The latter notion, which is what we will consider in this paper, means that reporting true values is always an optimal strategy for all agents, regardless of what the other agent does. Hence the notion that agents have incentive to act according to their true preferences is fully described by the DSIC property of a mechanism, which is formally defined as

 {Π(s,b)−s⋅A(s,b)≥Π(s′,b)−s⋅A(s′,b),b⋅A(s,b)−Π(s,b)≥b⋅A(s,b′)−Π(s′,b′)for all s,s′,b,b′∈R.

That is, both agents are always better off reporting their true preferences. It was shown by [hagerty] that such mechanisms are essentially fixed-price mechanisms, where the payment function is taken to be a (possibly random) single value , only depending on the distribution from which the valuations come from (and not on the valuations themselves). Then the allocation function is given by : trade occurs if and only if the seller values the item less than , and the buyer values it more.

#### Gains From Trade and Welfare.

As discussed above, we consider two benchmarks in this paper: gains from trade and welfare. The optimal gains from trade, given a distribution function from which the valuations are drawn, is defined as

 OPT-GFT(F)=E[1B>S(B−S)],

while the gains from trade achieved by a particular, non-random posted price is given by

 GFT(p,F)=E[1B≥p>S(B−S)].

Similarly, the corresponding welfare measures are defined as

 OPT-W(F) =E[S]+E[1B>S(B−S)]=E[S]+OPT-GFT(F), W(p,F) =E[S]+E[1B≥p>S(B−S)]=E[S]+GFT(p,F).

Some of our proofs in the next sections rely on explicit formulas for each of these quantities, which we now derive. The following derivation from [McAfee] gives a formula for the expected optimal gains from trade: letting denote the density of , we have

 OPT-GFT(F) =E[1B>S(B−S)] =∫∞0[(b−s)F(s)∣∣∣bs=0+∫b0F(s) ds]f(b) db =∫∞0∫b0F(s) ds⋅f(b) db =∫b0F(s) ds⋅[F(b)∣∣∣∞b=0−∫∞0F(b)2] db =∫∞0F(s) ds−∫∞0F(b)2 db =∫∞0F(x)(1−F(x)) dx.

A similar calculation gives an analogous formula for the gains from trade attained by posting a price Indeed, we have

 GFT(p,F) =E[1B≥p>S(B−S)] =E[1B≥p>S(B−p)]+E[1B≥p>S(p−S)] =E[1B≥p(B−p)]P(SS(p−S)]P(B≥p) =F(p)∫∞pf(b)(b−p) db+(1−F(p))∫p0f(s)(p−s) ds =F(p)∫∞p(1−F(x)) dx+(1−F(p))∫p0F(x) dx,

where the last equality is by integration by parts of each integral, as in the derivation of the formula for above.

## 3 Main Results

We organize the results into three subsections. In the first subsection, we focus on the setting where only one sample from is given, and show a -approximation of gains from trade and a -approximation of welfare. In the second subsection, we work in the setting of full access to the common distribution and show that posting the mean of as a price gives a -approximation of welfare, which is the best-possible in the symmetric model. Finally, in the third subsection we return to the asymmetric setting, where we now have different distributions and for the buyer and seller, respectively. In this setting we give a -approximation of welfare, showing that the mechanism of [blumrosendobzinski14] is not optimal; however, we show that is indeed optimal if one uses only the seller distribution to set the price.

### 3.1 Single-Sample Approximations of Gains From Trade and Welfare

We now show that our posted-price mechanism with one sample achieves a -approximation of the optimal gains from trade. As mentioned above, [Kang] showed that this is in fact the best-possible worst-case approximation in the symmetric setting, even among DSIC mechanisms with full access to the distribution We emphasize that this is not a bound, but an exact computation of the approximation ratio. Formally, since our mechanism posts a sample from as a price, we are interested in the ratio

 α:=Ep∼F[GFT(p,F)]OPT-GFT(F).

As we show next, we can compute exactly that independently of

###### Theorem 1.

The symmetric bilateral trade mechanism which under a valuation distribution posts a price achieves exactly 1/2 of the optimal gains from trade.

###### Proof.

We can write as

 ∫∞0[F(p)∫∞p(1−F(x)) dx+(1−F(p))∫p0F(x) dx]f(p) dp.

We now define and , which we will compute separately for simplicity, since we then have Let Now for the first term, we have

 γ1 =∫∞0f(p)F(p)∫∞p(1−F(x)) dx dp =∫∞0f(p)F(p)(λ−∫p0(1−F(x)) dx) dp =λ∫∞0f(p)F(p) dp−∫∞0f(p)F(p)∫p0(1−F(x)) dx dp =λ2−∫∞0f(p)F(p)∫p0(1−F(x)) dx dp =λ2−12F(p)2∫p0(1−F(x)) dx ∣∣∣∞0+12∫∞0F(p)2(1−F(p)) dp =12∫∞0F(p)2(1−F(p)) dp,

where we used that

 ∫∞0f(x)F(x) dx=12F(x)2 ∣∣∣∞0=1/2.

Note by integrating by parts. Thus for the second term we can write

 γ2 =lima→∞∫a0f(p)(1−F(p))∫p0F(x) dx dp =lima→∞[F(a)(1−12F(a))∫a0F(x) dx−∫a0F(p)2(1−12F(p)) dp].

Now for we have as Thus we have

 γ2 =lima→∞[θ∫a0F(x) dx−∫a0F(p)2(1−12F(p)) dp] =∫∞0[12F(p)−F(p)2(1−12F(p))] dp =12∫∞0F(p)(1−F(p))2 dp.

Putting things together we get

 Ep∼F[GFT(F;p)] =γ1+γ2 =12∫∞0F(p)2(1−F(p)) dp+12∫∞0F(p)(1−F(p))2 dp =12∫∞0F(p)(1−F(p)) dp.

Recalling that we get completing the proof. ∎

Given Theorem 1, it is simple to extend our result to obtain a 3/4 approximation of optimal welfare. Recall the representations of the welfare measures that we derived in the previous section:

 OPT-W(F) =μ+OPT-GFT(F), W(F,p) =μ+GFT(F,p),

where We can thus get the 3/4-approximation by applying Theorem 1:

###### Theorem 2.

The symmetric bilateral trade mechanism which under a valuation distribution posts a price achieves a 3/4-approximation of the optimal welfare.

###### Proof.

Using that we can write

 Ep∼F[W(F,p)]OPT-W(F) =μ+Ep∼F[GFT(F,p)]μ+OPT-GFT(F) =1−12OPT-GFT(F)μ+OPT-GFT(F) =1−lima→∞12∫a0F(x)(1−F(x)) dx∫a0xf(x) dx+∫a0F(x)(1−F(x)) dx =1−lima→∞12∫a0F(x) dx−∫a0F(x)2 dxaF(a)−∫a0F(x)2 dx.

But note that we have by the Cauchy–Schwarz inequality. Thus letting and noting that and we get

 ≤a1/2t−t2aF(a)−t2 ≤1F(a)t(a1/2−t)(a1/2+t)(a1/2−t) =1F(a)ta1/2+t. ≤1F(a)12 a→∞−−−→1/2.

Thus we have as desired. ∎

###### Remark 3.

Unlike our previous result for gains from trade, this approximation ratio is a lower bound, and not an exact computation. Indeed, in the case of welfare, the approximation ratio of our mechanism varies with the distribution Moreover, in further contrast with Theorem 1, this 3/4-approximation is not the best-possible among the class of truthful mechanisms with full knowledge of , since we show in the next subsection that the optimal ratio here is instead

Our remark notwithstanding, the approximation lower bound of 3/4 from Theorem 2 for our mechanism is tight, as we show next.

###### Lemma 4.

The 3/4-approximation bound of Theorem 2 is tight.

###### Proof.

Take the distribution functions parameterized by From the proof of Theorem 2, we get

 Ep∼F[W(Fr,p)]OPT%−W(Fr) =1−12∫10xr(1−xr)dxr∫10xrdx+∫10xr(1−xr)dx =1−121r+1−12r+1rr+1+1r+1−12r+1 r→0−−→1−1212=3/4,as % desired.\qed

### 3.2 Best-Possible Approximation of Welfare

We begin by showing that the optimal posted price in the symmetric setting is given simply by the mean of the common distribution.

###### Proposition 5.

The welfare-maximizing mechanism for symmetric bilateral trade is given by posting the mean of the common distribution :

 p∗=E[S]=E[B].
###### Proof.

For any positive price , we may rewrite the welfare function as

 W(p,F) =E[S]+E[(B−S)⋅1B>p≥p] =E[S]+E[B⋅1B>p]⋅F(p)−E[S⋅1S≤p]⋅[1−F(p)].

Because both agents’ values are drawn from the same distribution,

 E[B⋅1B>p]=E[S⋅1S>p]=E[S]−E[S⋅1S≤p].

Thus:

 W(p,F) =E[S]+E[B⋅1B>p]⋅F(p)−E[S⋅1S≤p]⋅[1−F(p)] =E[S]⋅[1+F(p)]−E[S⋅1S≤p] =E[S]⋅[1+F(p)]−p⋅F(p)+∫p0F(s) ds.

Now, for any fixed ,

 W(p±δ;F)=E[S]⋅[1+F(p±δ)]−(p±δ)⋅F(p±δ)+∫p±δ0F(x) dx.

We thus compute that

In the first expression, since is non-decreasing,

 ∫p∗p∗−δF(x) dx≥∫p∗p∗−δF(p∗−δ) dx=δ⋅F(p∗−δ).

Likewise, in the second expression, the same observation yields

 −∫p∗+δp∗F(x) dx≥−∫p∗+δp∗F(p∗+δ) dx=−δ⋅F(p∗+δ).

Therefore, with , substitution yields

Therefore the optimal fixed-price mechanism is given by . ∎

Proposition 5

shows that only the first moment of the agents’ distribution is required by the mechanism designer to determine the optimal fixed price; all higher moments are inconsequential. Therefore, precise knowledge of the agents’ distribution, other than the mean of the distribution, is irrelevant.

In general, the fixed-price mechanism

is not uniquely optimal. Suppose, for example, that each agent’s value is 0 with probability

and 1 with probability . Then any price is optimal. Nonetheless, under weak regularity conditions on , the mean is the uniquely optimal price.

###### Corollary 6.

Suppose that is differentiable in a neighborhood of its mean, so that its density is positive in that neighborhood. Then the optimal fixed-price mechanism that sets the price as the mean of the distribution is uniquely optimal.

The proof of Corollary 6 can be directly obtained from the proof of Proposition 5 by noting that the inequalities must hold strictly under the stated assumptions; we thus omit the details. In particular, Corollary 6 applies under the usual assumption that is continuously differentiable with positive density.

Despite Corollary 6, we proceed without additional regularity assumptions on to obtain a general analysis. Given the mean of the agents’ distribution, we can now determine the expected welfare that the optimal fixed-price mechanism achieves, :

 W(μ;F) =μ⋅[1+F(μ)]−E[S⋅1S≤μ] =μ+(μ−E[S|S≤μ])⋅F(μ).

An interesting observation here is that the maximum expected welfare achieved by any fixed-price mechanism depends on the agents’ distribution through only three quantities: (i) the mean of the distribution,

; (ii) the quantile of the mean,

; and (iii) the mean of the distribution conditional on being no greater than , . Consequently, any modification to that preserves these three quantities will lead to the same maximum expected welfare.

This observation motivates the following approach to find the approximation ratio we achieve by posting the mean as a price which, as we showed, is the best-possible among DSIC mechanisms:

 infFW(μF;F)OPT-W(F)=infFsupp∈R+W(p,F)OPT-W(F). (A)

Define the subspace of probability distributions that fixes the three quantities from above:

 ΔL1(R+;μ,μ1,γ):={F∈ΔL1(R+):E[S]=μ,E[S|S≤μ]=μ1,F(μ)=γ}.

To compute (A), we first compute

The original problem (A) is thus equivalent to

The objective of this decoupling is to reduce the dimension of our optimization problem. Instead of minimizing over the infinite-dimensional space , our outer optimization problem is simplified to minimization over three variables. However, the inner optimization problem (A’) still requires minimizing over the infinite-dimensional space . To reduce the dimension of (A’), define the space of distributions supported on at most 4 points that fixes the three quantities:

 Δ(4)L1(R+;μ,μ1,γ):={F∈ΔL1(R+;μ,μ1,γ):F(x)= q0⋅1x≥0+q1⋅1x≥x1 +q2⋅1x≥x2+q3⋅1x≥1}.

The probability masses are given by .

###### Lemma 7.

For any fixed , and ,

We defer the proof of Lemma 7 to Appendix A and sketch the argument here. Fix . We may assume without loss of generality that is a finite support discrete distribution because any can be approximated arbitrarily well by a corresponding distribution in supported on finitely many points. Moreover, by rescaling if necessary, we may also assume that .

To solve the inner problem (A’), our earlier observation shows that, given , and , is constant for any . That is:

Thus (A’) can be solved by maximizing over all . Note that, from the definition,

 OPT-W(F)=E[max{S,B}]=E[S]+12E[|B−S|].

Therefore, can be optimized by the following procedure:

• For any probability mass in , split the mass into two equal masses. Move each mass in opposite directions, until one mass hits the boundary of the interval .

• For any probability mass in , split the mass into two equal masses. Move each mass in opposite directions, until one mass hits the boundary of the interval , for some sufficiently small .222Note that this is required so that this operation does not change .

Each of these operations is mean-preserving in the intervals and , so neither changes the quantities , and . Therefore, the operations preserve the maximum expected welfare achieved by any fixed-price mechanism, . However, they increase in the respective intervals, and so the operations increase . Applying these operations recursively,333In the proof of Lemma 7 in Appendix A, we consider only the limiting distribution obtained from recursive application of these operations; however, these operations motivate the construction of in the proof. we obtain a 4-point distribution such that

 W(p;~F)OPT-W(~F)≤W(p,F)OPT-W(F).

Since , this proves the result of Lemma 7.

As a consequence of Lemma 7, we can restrict attention to 4-point distributions when solving the minimax problem (A). Such distributions can be written as444The definition of is slightly more general as it allows for any in the form

However, from the proof sketch of Lemma 7 (and further justified in the proof given in Appendix A), we may narrow our attention to distributions where and for sufficiently small .

 F(x)=q0⋅1x≥0+q1⋅1x≥μ+q2⋅1x≥μ+δ+q3⋅1x≥1for some δ∈(0,1−μ).

Here, the probability masses , , and satisfy and the three additional conditions:

Since , and are given, we can rewrite as follows:555Details of this computation can be found in the proof of Lemma 7 given in Appendix A.

 F(x)=γ(1−μ1μ)⋅1x≥0+μ1γμ⋅1x≥μ+1−γ−μ+μ1γ1−μ−δ⋅1x≥μ+δ+μγ−μ1γ−δ−δγ1−μ−δ⋅1x≥1.

This parametrization of allows us to easily compute and . We have shown previously that depends only on , and :

 W(μ;F)=μ+(μ−μ1)⋅γ.

On the other hand, can be computed by observing that must be either: (i) both no greater than ; (ii) both greater than ; or (iii) on different sides of . Thus:

 OPT-W(F) =E[max{B,S}] =E[max{B,S}⋅1B,S≤μ]+E[max{B,S}⋅1B,S>μ] +E[max{B,S}⋅1max{B,S}>μ1min{B,S}≤μ].

The first and third contributions can be computed separately:

The second contribution can be bounded above independently of :

 E[max{B,S}⋅1B,S>μ] =[1−F(μ)]2⋅{1−(1−μ−δ)⋅P[B=S=μ+δ|B,S>μ]} =(1−γ)2−(1−γ−μ+μ1γ)21−μ−δ ≤(1−γ)2−(1−γ−μ+μ1γ)21−μ.

Consequently, we obtain an upper bound for that is independent of :

Therefore, we have

 W(μ;F)OPT-W(F)≥μ+(μ−μ1)γμ1γ2(2−μ1μ)+2γ(μ−γμ1)+(1−γ)2−(1−γ−μ+μ1γ)21−μ.

This allows us to bound the value of the minimax problem (A) from below:

The lower bound in the above expression can be computed. Notably, optimization is carried out over a 3-dimensional space, and so elementary techniques apply. While we relegate the computational details to Appendix A, we state the result here:

###### Lemma 8.

The minimax value of the game is bounded below: