We consider an auction design problem under network flow constraints. We focus on pricing mechanisms that provide fair solutions, where fairness is defined in absolute and relative terms. The absolute fairness is equivalent to “no individual losses” assumption. The relative fairness can be verbalized
as follows: no agent can be treated worse than any other in similar circumstances. Ensuring the fairness conditions makes only part of the social welfare available in the auction to be distributed on pure market rules. The rest of welfare must be distributed without market rules and constitutes the so-called

Classical auction (through the whole paper by “auction” we mean closed double sealed exchange-like mechanism; in other words an “auction” is a set of trading rules for an exchange) mechanisms are based on the supply/demand curves intersection which sets accepted and rejected offers and determines the uniform market price. However, in many real-world infrastructure economies, a commodity flow is limited by the resources of some network system. This leads to a concept of the networked auctions [

In [

To address fairness in the electricity markets, the locational marginal pricing (LMP) was introduced by Schweppe [

In this paper we introduce the

The paper is organised as follows. A discussion of related works is provided in Section

The concept of fairness plays an important role in resource allocation problems [

Unfortunately, all of these concepts of fairness and their underlying axiomatisation cannot be applied directly when the allocation is performed via auction mechanism. In this case, the fairness concept must take into consideration the prices of bids and thus different utility of bidders. Moreover, the incompleteness of information possessed by each bidder is also important.

Auction has been widely considered as a mechanism of resource allocation in selfish, multiagents environments [

Murillo et al. [

Wu et al. [

The concept of fairness for multicommodity auctions has been formalized by Toczyłowski in [

The term

There are also some studies, that refer to the PoF indirectly; They consider loss of efficiency due to fair conditions, mainly in allocation problems. In most works, the price of fairness or similar notion is defined as the performance loss incurred relative to utility, in making allocations under one of several possible fairness criteria formulations. In [

In [

The

We consider an organized market in which the sellers and buyers submit their offers. Then, the auction mechanism is run to find the winning offers and to set the value flows. The auction rules can be divided into two steps: the winners determination and the pricing.

Let us assume that an infrastructure network is modeled by graph

The sellers submit the set of offers

The maximal total social welfare

The value mechanism is responsible for the surplus distribution. The distribution is usually expressed with the use of payment information

A very generic approach to modeling the value mechanisms was formulated by Toczyłowski and called

Model of pricing with nodal prices

Let us assume that there is a set of

For a given pricing mechanism it is interesting whether it is nondominated solution of multicriteria problem (

For a given set of nodal prices

Two notions of fairness have been introduced in [

Fairness in absolute sense means that no offer brings individual loss. Figure

Cost

Similarly, the compensation of competitive buyer losses should be introduced. For forced buy offer the compensation is

Mechanism is

Loss of profits

Also, two participants connected with undersaturated edge should have the same nodal prices. If this would not be satisfied, then the entity with the worse nodal price could trade in the neighbor’s node with better profits.

A mechanism

Let

The difference between total social welfare and welfare distributed under the pure market rules is a price that must be paid to maintain fairness [

Let

The space

Moreover, for the sake of generality, we also consider the mechanisms that, unlike the LMP or classical uniform pricing, assume two nodal market prices at each node: buying price

In the perfect situation, when no congestion manifests in the network, the maximal economic benefit

Price of fairness can be expressed as follows:

Market sell cost received by the sellers is

Market buy cost received by the buyers is

The basic balance is as follows:

Let us analyze the properties of

Similar analysis can be done for function

Functions

Market sell cost and market buy cost functions.

The PoF is a convex function with respect to the value of market trade

The proof of convexity of function

Notice that above property is true for all values of

Now, we will show that the minimum of PoF is reached for market price

The PoF for market price

We assume that congestions involve some reduction of accepted volume of offers from set

The above expression can be also rewritten in the following form:

Decreasing the market price causes increase in costs of forced sell with value

Similar reasoning can be carried out for the buyers, showing that decreasing

The PoF for market price

The proof is similar to the proof of the previous lemma. Let us assume that congestions cause the reduction of accepted volume of offers from the set

The increase of market price causes the decrease cost of forced sell by

Total change in PoF of balancing is as follows:

Finally, we can formulate the theorem about the minimal PoF.

The minimum of PoF is achieved at the market price

The proof is clear on the basis of Lemma

In the paper, we have introduced and analyzed the concept of

Our results open the doors for further investigations of new mechanisms. We have shown that there is still a lot of space for new mechanisms that can be even better than locational marginal pricing, at least in some criteria, for example, PoF. We have introduced the space of mechanism

The author declares that there is no conflict of interests regarding the publication of this paper.