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In the bilateral trading model developed by Myerson [1], an ex post efficient mechanism that satisfies incentive compatibility and individual rationality has been proved to be not existed. This paper is aimed at the discussion of modified bilateral trading model where the buyer and seller’s valuations for the object are assumed to be not independent. Under such assumption, we concern about the existence of an ex post efficient mechanism satisfying individual compatible and individual rational conditions. We will give a necessary condition of such mechanism, not precisely though, to identify the existence. At the end of this paper, we provide some empirical evidences about the non-independent assumption and suggest some directions for further research, both empirical and theoretical.

Uncertainty and asymmetric information have been under central discussion in mechanism design and auction theories. Many theories have been put forward to model the behavior of market or mechanism under asymmetric information, which can be traced back to the research in Lemon market and adverse selection of job market [

In Vickrey-Clarke-Groves (VCG) auction mechanism design, the optimal auction mechanism is clear for a class of distribution where agents types are independently and identically distributed, which has been proved in references [

Previous research about bilateral trading problem ( [_{1} and v_{2} are independent. The assumption directly implies that the trading area is a rectangle [ a 1 , b 1 ] × [ a 2 , b 2 ] . The proof of the negative expected utility of the point ( b 1 , a 2 ) indicates that even a mechanism satisfies the incentive compatible and ex post efficient, it would fail to meet each player’s individual rationality. So the market can only support an individual rational, incentive compatible and ex post efficient mechanism with strictly positive subsidy.

We can generalize the model by relaxing this assumption. Now the random variables v_{1} and v_{2} are distributed on a convex area D. So if we can prove that each point (x, y) in area D yields positive utility for each player, i.e. U 1 ( x ) + U 2 ( y ) > 0 , we can prove the existence of the corresponding mechanism. This paper gives theoretical analyses about the generalized trading model, identifying the existence of such a mechanism.

This paper is organized in the following pattern: in Section 2, we present the basic definition of the modified model, define some basic parameters for further analysis. Part 3 repeated the analysis that is common for a principal-agent model under the changing in distribution. The expressions of incentive compatible and individual rational are slightly different. In Section 4, we argued our main result for generalized bilateral trading model: there may exist an individual rational, incentive compatible and ex post efficient mechanism when the distribution of the type of each player is not independent anymore. Furthermore, in Section 5, we explained the reasons that guarantee the possibility of the existence of such generalized case in an empirical way. The conclusion is shown in part 6.

Consider a trading problem where individual 1 is a buyer and individual 2 is a seller. Seller owns an object that buyer wants. Each individual has his own valuation about the object.

Denote the seller’s valuation of the object: v_{1}, the buyer’s valuation of the object: v_{2}. v_{1}, v_{2} are random variables distributed on a convex differentiable zone D with density function f ( v 1 , v 2 ) . The utility of the seller is v_{2}-x and the utility of the buyer is x-v_{1} if the mechanism specifies x as the executing price.

According to an essential result of bargain game called revelation principle, a mechanism can be determined by two functions: x ( v 1 , v 2 ) , p ( v 1 , v 2 ) if the mechanism is a direct bargaining mechanism.

x ( v 1 , v 2 ) : the price of the object if buyer has valuation v_{1} and the seller has valuation v_{2}.

p ( v 1 , v 2 ) : the possibility of the occurrence of trade if buyer and seller’s valuation are v_{1} and v_{2} respectively.

The difference lies in the definition domain of x ( v 1 , v 2 ) , p ( v 1 , v 2 ) is now on the area D.

Given any equilibrium of any bargain game, there exist an equivalent incentive-compatible direct mechanism by first asking the buyer and seller each to confidentially report his valuation, then determine the corresponding x ( v 1 , v 2 ) , p ( v 1 , v 2 ) by computing according to original mechanism. When we are analyzing bilateral trading mechanism, we only consider such direct mechanism.

We need to define some new variables in order to get the expected utility of buyer and seller. But before giving the specific definition, we need to do some technical work.

Suppose the smallest rectangle R containing D is [ a 1 , b 1 ] × [ a 2 , b 2 ] . The following figure shows an example of such a rectangle. Then we extend the definition domain of f ( v 1 , v 2 ) , p ( v 1 , v 2 ) and x ( v 1 , v 2 ) to [ a 1 , b 1 ] × [ a 2 , b 2 ] . The

specific method is: If ( v 1 , v 2 ) ∈ D , then f, p, x take original value. If ( v 1 , v 2 ) ∉ D , then f, p, x equal to 0. We use the extended f, p, x in the following discussion.

As is shown in

We denote the red curve as L, which is a part of D’s boundary and is tangent to line: y = a 2 and line: x = b 1 .

After the above technical works, the buyer and seller’s conditional expectation of x, p and their final utility can be written as:

x ¯ 1 ( v 1 ) = ∫ a 2 b 2 x ( v 1 , t 2 ) f ( v 1 , t 2 ) d t 2 h 1 ( v 1 ) , x ¯ 2 ( v 2 ) = ∫ a 1 b 1 x ( t 1 , v 2 ) f ( t 1 , v 2 ) d t 1 h 2 ( v 2 ) ,

p ¯ 1 ( v 1 ) = ∫ a 2 b 2 p ( v 1 , t 2 ) f ( v 1 , t 2 ) d t 2 h 1 ( v 1 ) , p ¯ 2 ( v 2 ) = ∫ a 1 b 1 p ( t 1 , v 2 ) f ( t 1 , v 2 ) d t 1 h 2 ( v 2 ) ,

U 1 ( v 1 ) = x ¯ 1 ( v 1 ) − v 1 p ¯ 1 ( v 1 ) , U 2 ( v 2 ) = v 2 p ¯ 2 ( v 2 ) − x ¯ 2 ( v 2 ) ,

h 1 ( v 1 ) = ∫ a 2 b 2 f ( v 1 , t ) d t , h 2 ( v 2 ) = ∫ a 1 b 1 f ( t , v 2 ) d t .

The U_{1}, U_{2} defined above is the expected total utility gain from the trade. Thus their economic meanings are well defined. We may call h 1 ( v 1 ) , h 2 ( v 2 ) : the generalized density function of each individual.

We first review the definition of incentive compatibility and individual rationality.

Incentive compatibility:

A mechanism (p, x) is individual compatible iff for any v_{1} and v ′ 1 in [ a 1 , b 1 ]

U 1 ( v 1 ) ≥ x ¯ 1 ( v ′ 1 ) − v 1 p ¯ 1 ( v ′ 1 )

And for any v_{2} and v ′ 2 in [ a 2 , b 2 ]

U 2 ( v 2 ) ≥ v 2 p ¯ 2 ( v ′ 2 ) − x ¯ 2 ( v ′ 2 )

The economic interpretation of incentive compatibility is that each of the players can maximize their expected utility by reporting their type honestly, rather than telling a lie.

We can derive that any incentive compatible mechanism must satisfy:

U 1 ( v 1 ) = U 1 ( b 1 ) + ∫ v 1 b 1 p ¯ 1 ( t 1 ) d t 1 , U 2 ( v 2 ) = U 1 ( a 2 ) + ∫ a 2 v 2 p ¯ 2 ( t 2 ) d t 2

The above relations are natural expansion of the classic results developed in optimal auction design and VCG mechanism design [

Individual rationality:

Also known as participating constraint, Individual rationality requires that, in a game, each player has positive expected utility:

U 1 ( x ) + U 2 ( y ) ≥ 0 for any (x, y) in D.

If a mechanism is individual rational, players will obtain non-negative expected utility by participating the game.

Here is the main result of the generalized model:

Theorem:

For any incentive compatible mechanism in generalized bilateral trading model. The probability function p ( v 1 , v 2 ) satisfies p ¯ 1 ( v 1 ) is decreasing on v_{1}, p ¯ 2 ( v 2 ) is increasing on v_{2}. Moreover, the mechanism satisfies individual rationality condition iff for any point (x, y) on the boundary of D. U 1 ( x ) + U 2 ( y ) ≥ 0 . At last, for any point (x, y) on L, we have:

U 1 ( x ) + U 2 ( y ) = ∬ D ( v 2 − v 1 ) f ( v 1 , v 2 ) p ( v 1 , v 2 ) d v 1 d v 2 − ∫ a 1 b 1 H 1 ( t 1 ) p ¯ 1 ( t 1 ) d t 1 − ∫ a 2 b 2 ( 1 − H 2 ( t 2 ) ) p ¯ 2 ( t 2 ) d t 2 + E 1 + E 2

And we have the definition of H 1 , H 2 , E 1 , E 2 :

H 1 ( t ) = ∫ a 1 t h 1 ( x ) d x , H 2 ( t ) = ∫ a 2 t h 2 ( x ) d x

E 1 = ∫ x b 1 p ¯ 1 ( t 1 ) d t 1 , E 2 = ∫ a 2 y p ¯ 2 ( t 2 ) 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:

U 1 ( x ) + U 2 ( y ) = ∬ D ( v 2 − v 1 ) f ( v 1 , v 2 ) p ( v 1 , v 2 ) d v 1 d v 2 − ∫ a 1 b 1 H 1 ( t 1 ) p ¯ 1 ( t 1 ) d t 1 − ∫ a 2 b 2 ( 1 − H 2 ( t 2 ) ) p ¯ 2 ( t 2 ) d t 2 + E 1 + E 2 ≥ 0

Due to the monotonic proposition of p ¯ 1 ( v 1 ) , p ¯ 2 ( v 2 ) , for any point (m, n) in area D, there exist a point (x, y) on L s.t.

U 1 ( x ) + U 2 ( y ) ≥ U 1 ( m ) + U 2 ( n ) . Thus the deduction holds.

Tips: The proof of theorem 1 is again similar to the proof of Myerson [

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 ( v 1 , v 2 ) = { 1 if v 1 ≤ v 2 ; 0 if v 1 > v 2 .

It has been proved that for the independent distributed valuation model developed in Myerson [

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 ( x ) + U 2 ( y ) ≥ 0 . So we need to find the set of (x, y) that minimize U 1 ( x ) + U 2 ( y ) .

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

[ x 2 , b 1 ] × [ a 2 , x 1 ] , denoting S.

∬ D ( v 2 − v 1 ) f ( v 1 , v 2 ) p ( v 1 , v 2 ) d v 1 d v 2 − ∫ a 1 b 1 H 1 ( t 1 ) p ¯ 1 ( t 1 ) d t 1 is always negative, which contradicts individual rational condition (

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:

∬ D ( v 2 − v 1 ) f ( v 1 , v 2 ) p ( v 1 , v 2 ) d v 1 d v 2 − ∫ a 1 b 1 H 1 ( t 1 ) p ¯ 1 ( t 1 ) 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 ( x ) + U 2 ( y ) .

Take derivation of U 1 ( x ) + U 2 ( y ) 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:

− p ¯ 1 ( x ) d x + p ¯ 2 ( y ) d y = 0

which can be also written as:

p ¯ 1 ( x ) p ¯ 2 ( y ) = d y d x

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

Notice that p ¯ 1 ( x ) is decreasing on x, while p ¯ 2 ( y ) is increasing on y, and the shape of L ensures that y is increasing on x, thus p ¯ 1 ( x ) p ¯ 2 ( y ) is decreasing on x.

Also notice that d y d x is increasing on x since D is convex. So p ¯ 1 ( x ) p ¯ 2 ( y ) − d y 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 ( x ) + U 2 ( y ) 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 [

In the case shown in

On D, the density function has form:

f ( v 1 , v 2 ) | D = f 1 ( v 1 ) × f 2 (v2)

Also, denote the corresponding “distribution” functions (not strict distribution functions because they don’t satisfy: F i ( ∞ ) = 1 ) as F 1 ( v 1 ) , F 2 ( v 2 ) as the integral function of f 1 ( v 1 ) and f 2 ( v 2 ) , 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 = ∬ D f ( v 1 , v 2 ) d v 1 d v 2

g ( v 1 , v 2 ) = f ( v 1 , v 2 ) | S

G 2 ( v 2 ) = ∫ a 2 v 2 g 2 ( x ) d x , G 1 ( v 1 ) = ∫ a 1 v 1 g 1 ( x ) d x

g 1 ( v 1 ) = ∫ a 2 c 2 g ( v 1 , t ) d t , g 2 ( v 2 ) = ∫ a 1 c 1 g ( v 2 , t ) d t

If we assume c 1 is so small such that the minimum value of ( U 1 ( x ) + U 2 ( y ) ) achieves at U 1 ( b 1 ) + U 2 ( c 2 ) .

Then we can obtain the minimum utility:

min ( U 1 ( x ) + U 2 ( y ) ) = U 1 ( b 1 ) + U 2 ( c 2 ) = − r ∫ c 2 b 1 ( 1 − F 2 ( t ) ) F 1 ( t ) d t + ∬ S G 2 ( v 2 ) g 2 ( v 2 ) − G 1 ( v 1 ) g 1 ( v 1 ) p ( v 1 , v 2 ) g ( v 1 , v 2 ) d v 1 d v 2

Minimum utility can be positive if r is very small and G 2 ( v 2 ) g 2 ( v 2 ) − G 1 ( v 1 ) g 1 ( v 1 ) is

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.

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 ( v 1 , v 2 ) by sampling and calculate the minimum value of ( U 1 ( x ) + U 2 ( y ) ) 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.

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.

Tao, R.C. (2018) A Generalized Bilateral Trading Model. Open Journal of Business and Management, 6, 606-615. https://doi.org/10.4236/ojbm.2018.63046