${E}_{1}={\displaystyle {\int}_{x}^{{b}_{1}}{\stackrel{\xaf}{p}}_{1}\left({t}_{1}\right)\text{d}{t}_{1}},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{E}_{2}={\displaystyle {\int}_{{a}_{2}}^{y}{\stackrel{\xaf}{p}}_{2}\left({t}_{2}\right)\text{d}{t}_{2}}$

A simple deduction of the theorem is: if a mechanism is individual rational, then for any point (x, y) on the line L:

$\begin{array}{l}{U}_{1}\left(x\right)+{U}_{2}\left(y\right)\\ ={\displaystyle \underset{D}{\iint}\left({v}_{2}-{v}_{1}\right)f\left({v}_{1},{v}_{2}\right)p\left({v}_{1},{v}_{2}\right)\text{d}{v}_{1}\text{d}{v}_{2}}-{\displaystyle {\int}_{{a}_{1}}^{{b}_{1}}{H}_{1}\left({t}_{1}\right){\stackrel{\xaf}{p}}_{1}\left({t}_{1}\right)\text{d}{t}_{1}}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.05em}}-{\displaystyle {\int}_{{a}_{2}}^{{b}_{2}}\left(1-{H}_{2}\left({t}_{2}\right)\right){\stackrel{\xaf}{p}}_{2}\left({t}_{2}\right)\text{d}{t}_{2}}+{E}_{1}+{E}_{2}\ge 0\end{array}$

Due to the monotonic proposition of ${\stackrel{\xaf}{p}}_{1}\left({v}_{1}\right),{\stackrel{\xaf}{p}}_{2}\left({v}_{2}\right)$ , for any point (m, n) in area D, there exist a point (x, y) on L s.t.

${U}_{1}\left(x\right)+{U}_{2}\left(y\right)\ge {U}_{1}\left(m\right)+{U}_{2}\left(n\right)$ . Thus the deduction holds.

Tips: The proof of theorem 1 is again similar to the proof of Myerson [1] Theorem 1. So we don’t present the detailed proof of theorem 1 here.

4. The Existence of an Ex Post Efficient Mechanism

Indeed, an incentive compatible and individual rational mechanism is desirable for practical usage. But in order to consider the efficiency of a mechanism, we need to consider one more property, which is ex post efficiency. In general, even if a mechanism is incentive compatible and individual rational, the inefficient case where the valuation ${v}_{1}>{v}_{2}$ but the mechanism still tells that a trade should be made may appear, which will cause each player becoming worse off.

First, we define another concept that may help us to measure the efficiency of a mechanism.

A mechanism is ex post efficient if and only if

$p\left({v}_{1},{v}_{2}\right)=\{\begin{array}{l}1\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.05em}}\text{if}\text{\hspace{0.17em}}{v}_{1}\le {v}_{2};\\ 0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{if}\text{\hspace{0.17em}}{v}_{1}>{v}_{2}.\end{array}$

It has been proved that for the independent distributed valuation model developed in Myerson [1] , if a mechanism is incentive compatible and ex post efficient. It must fail to satisfy individual rationality. Now we reconsider this problem in generalized model.

The main difficulty of making a quick justification of the existence of an incentive compatible, individual rational, and ex post efficient mechanism because we need to make sure that any point (x, y) on L, satisfies ${U}_{1}\left(x\right)+{U}_{2}\left(y\right)\ge 0$ . So we need to find the set of (x, y) that minimize ${U}_{1}\left(x\right)+{U}_{2}\left(y\right)$ .

A special subarea of D raised our attention. Suppose line: y=x intersect the boundary of area D at $\left({x}_{1},{x}_{1}\right),\left({x}_{2},{x}_{2}\right)$ and without loss of generality, we suppose ${x}_{1}<{x}_{2}$ . The special area if D is the intersection of D and domain

$\left[{x}_{2},{b}_{1}\right]\times \left[{a}_{2},{x}_{1}\right]$ , denoting S.

$\iint}_{D}\left({v}_{2}-{v}_{1}\right)f\left({v}_{1},{v}_{2}\right)p\left({v}_{1},{v}_{2}\right)\text{d}{v}_{1}\text{d}{v}_{2}}-{\displaystyle {\int}_{{a}_{1}}^{{b}_{1}}{H}_{1}\left({t}_{1}\right){\stackrel{\xaf}{p}}_{1}\left({t}_{1}\right)\text{d}{t}_{1$ is always negative, which contradicts individual rational condition (Figure 2).

If S is not empty set, then there is no incentive compatible, individual rational, and ex post efficient mechanism simply because ${E}_{1}+{E}_{2}$ is always zero. In fact, we can assure that:

$\iint}_{D}\left({v}_{2}-{v}_{1}\right)f\left({v}_{1},{v}_{2}\right)p\left({v}_{1},{v}_{2}\right)\text{d}{v}_{1}\text{d}{v}_{2}}-{\displaystyle {\int}_{{a}_{1}}^{{b}_{1}}{H}_{1}\left({t}_{1}\right){\stackrel{\xaf}{p}}_{1}\left({t}_{1}\right)\text{d}{t}_{1$ is always negative, which contradicts individual rational condition.

We have identified a necessary condition for the existence of an incentive compatible, individual rational, and ex post efficient mechanism, but we also need some more accurate method to estimate the value of ${U}_{1}\left(x\right)+{U}_{2}\left(y\right)$ .

Take derivation of ${U}_{1}\left(x\right)+{U}_{2}\left(y\right)$ with respect to x and y. we can get the following equation. This relies on the assumption that D is a differentiable area.

We consider the first order condition:

$-{\stackrel{\xaf}{p}}_{1}\left(x\right)\text{d}x+{\stackrel{\xaf}{p}}_{2}\left(y\right)\text{d}y=0$

which can be also written as:

$\frac{{\stackrel{\xaf}{p}}_{1}\left(x\right)}{{\stackrel{\xaf}{p}}_{2}\left(y\right)}=\frac{\text{d}y}{\text{d}x}$

Figure 2. The case where S is not empty set.

Next we will prove that the first order condition has one and only one solution:

Notice that ${\stackrel{\xaf}{p}}_{1}\left(x\right)$ is decreasing on x, while ${\stackrel{\xaf}{p}}_{2}\left(y\right)$ is increasing on y, and the shape of L ensures that y is increasing on x, thus $\frac{{\stackrel{\xaf}{p}}_{1}\left(x\right)}{{\stackrel{\xaf}{p}}_{2}\left(y\right)}$ is decreasing on x.

Also notice that $\frac{\text{d}y}{\text{d}x}$ is increasing on x since D is convex. So $\frac{{\stackrel{\xaf}{p}}_{1}\left(x\right)}{{\stackrel{\xaf}{p}}_{2}\left(y\right)}-\frac{\text{d}y}{\text{d}x}$

must have only one zero point and the solution is a minimum value point. So, given a distribution of V_{1} and V_{2}, we can numerically calculate the minimum
${U}_{1}\left(x\right)+{U}_{2}\left(y\right)$ and judge the existence of an incentive compatible, individual rational, and ex-post efficient mechanism by the sign of the minimum value.

This time, we cannot prove the non-existence theorem in reference [1] and [9] , because we can easily construct a special case to prove its existence. Although the area doesn’t satisfy the convex and differentiable assumption, the example is indeed the simplest way to construct such a special case. Since the convex and differentiable assumptions are not substantially essential to our main result, we can release these assumptions temporary.

In the case shown in Figure 3, the whole area D consists of two rectangles: D and S. $D:\left[{a}_{1},{b}_{1}\right]\times \left[{a}_{2},{b}_{2}\right]$ , $S:\left[{a}_{1},{c}_{1}\right]\times \left[{a}_{2},{c}_{2}\right]$ and ${c}_{2}<{b}_{1}.$

On D, the density function has form:

${f\left({v}_{1},{v}_{2}\right)|}_{D}={f}_{1}\left({v}_{1}\right)\times {f}_{2}(v2)$

Also, denote the corresponding “distribution” functions (not strict distribution functions because they don’t satisfy:
${F}_{i}\left(\infty \right)=1$ ) as
${F}_{1}\left({v}_{1}\right),{F}_{2}\left({v}_{2}\right)$ as the integral function of
${f}_{1}\left({v}_{1}\right)$ and
${f}_{2}\left({v}_{2}\right)$ , which implies V_{1} and V_{2} are “locally” independent on D.

We need some new notations to state the special case in order.

$r={\displaystyle \underset{D}{\iint}f\left({v}_{1},{v}_{2}\right)\text{d}{v}_{1}\text{d}{v}_{2}}$

$g\left({v}_{1},{v}_{2}\right)={f\left({v}_{1},{v}_{2}\right)|}_{S}$

${G}_{2}\left({v}_{2}\right)={\displaystyle {\int}_{{a}_{2}}^{{v}_{2}}{g}_{2}\left(x\right)\text{d}x},\text{\hspace{0.17em}}\text{\hspace{0.17em}}{G}_{1}\left({v}_{1}\right)={\displaystyle {\int}_{{a}_{1}}^{{v}_{1}}{g}_{1}\left(x\right)\text{d}x}$

${g}_{1}\left({v}_{1}\right)={\displaystyle {\int}_{{a}_{2}}^{{c}_{2}}g\left({v}_{1},t\right)\text{d}t},\text{\hspace{0.17em}}\text{\hspace{0.17em}}{g}_{2}\left({v}_{2}\right)={\displaystyle {\int}_{{a}_{1}}^{{c}_{1}}g\left({v}_{2},t\right)\text{d}t}$

If we assume ${c}_{1}$ is so small such that the minimum value of $\left({U}_{1}\left(x\right)+{U}_{2}\left(y\right)\right)$ achieves at ${U}_{1}\left({b}_{1}\right)+{U}_{2}\left({c}_{2}\right)$ .

Then we can obtain the minimum utility:

$\begin{array}{l}\mathrm{min}\left({U}_{1}\left(x\right)+{U}_{2}\left(y\right)\right)={U}_{1}\left({b}_{1}\right)+{U}_{2}\left({c}_{2}\right)\\ =-r{\displaystyle {\int}_{{c}_{2}}^{{b}_{1}}\left(1-{F}_{2}\left(t\right)\right){F}_{1}\left(t\right)\text{d}t}+{\displaystyle \underset{S}{\iint}\frac{{G}_{2}\left({v}_{2}\right)}{{g}_{2}\left({v}_{2}\right)}-\frac{{G}_{1}\left({v}_{1}\right)}{{g}_{1}\left({v}_{1}\right)}p\left({v}_{1},{v}_{2}\right)g\left({v}_{1},{v}_{2}\right)\text{d}{v}_{1}\text{d}{v}_{2}}\end{array}$

Minimum utility can be positive if r is very small and $\frac{{G}_{2}\left({v}_{2}\right)}{{g}_{2}\left({v}_{2}\right)}-\frac{{G}_{1}\left({v}_{1}\right)}{{g}_{1}\left({v}_{1}\right)}$ is

Figure 3. The simplified case where an IC IR EPE mechanism may exist.

large enough. These conditions can be satisfied if we intend to construct such a density function.

We have proved that under generalized bilateral trading model, there may exist an incentive-compatible individual rational and ex-post efficient mechanism. Anyway, the economic intuition about the generalized model should also be declared. Still, some empirical and further theoretical analyses are also needed to be made.

5. Further Interpretations

After the proof of the possible existence of an incentive-compatible individual rational and ex-post efficient mechanism in the generalized bilateral trading model, now we turn our attention to the empirical explanation to the key assumption, which is dealing with the distribution of the agents’ type.

Empirically, since beliefs of people vary from person to person, different types of individual also face different choice of their seller in the case of bilateral trading. That is a possible explanation for our modified assumptions. A typical example is: if you want to buy a car, you can buy it at second-hand car market but you can also buy it at flagship store. It’s an often case that wealthy consumers buy their cars at flagship store while the poor not. This phenomenon results in that different cohorts are matched if consumers don’t share the same type (willingness-to-pay). Especially, When assuming there are a large number of such markets. The trading area will be restricted to the generalized trading model, where types are not independent by the updated belief or rational expectation.

There is another question: if we can buy the same thing at various market. It will be a dominant strategy to choose a market that has relative low expected valuation of the object. Under this assumption, the high-end market will crash and all buyers will purchase goods at low-end market or second-hand market, while this is obviously not the fact.

The generalized market may have an incentive compatible, individual rational and ex post efficient mechanism theoretically, but we can further analysis the empirical data to decide whether there exists an incentive compatible, individual rational and ex post efficient mechanism for each market.

Empirically, we can study the bilateral trading results in various markets to provide supporting evidences for the generalized model. We should first specify how many kinds of markets exist in all and examine the trading data of each market to obtain an unbiased estimation of area D. Then we find a best differentiable area D’ that can be treated as an approximation of D.

Then we need to estimate the corresponding density function $f\left({v}_{1},{v}_{2}\right)$ by sampling and calculate the minimum value of $\left({U}_{1}\left(x\right)+{U}_{2}\left(y\right)\right)$ by finding the point satisfying first order condition on the boundary of D’. Using the above result, we can identify whether an incentive compatible, individual rational, ex post efficient mechanism can exist.

Theoretically, finding the sufficient and necessary conditions of the existence of an incentive compatible, individual rational, and ex post efficient mechanism is still an unfinished work.

6. Conclusion

In conclusion, this paper successfully generalizes the bilateral trading model by easing the restriction on the distribution, which now can be non-independent. The two main changes brought by the modification are: 1) the expression of the expected utility with respect to each player; 2) the changes in determining condition for individual rationality. These two changes make it possible for the existence of an incentive compatible, individual rational, and ex post efficient mechanism.

Conflicts of Interest

The authors declare no conflicts of interest.

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