1 Introduction
How should sellers price their items in a market? This is arguably the central question in any market setting. The emergence of electronic markets has significantly changed the nature of markets. On the one hand, information collected online has made it possible for vendors to have detailed information on potential buyers. On the other hand, buyers now have easy access to the variety of items different sellers have available and their prices. Buyers are interested in purchasing a bundle of items maximizing their utility for the items purchased. Sellers are interested in maximizing their profit. Furthermore, the size and speed of online markets necessitates sellers to set their prices competitively and quickly.
Following Babaioff, Nisan and Paes Leme [2], we will view online markets as a full information game in which the sellers are the strategic agents. We are therefore interested in pricing equilibria in combinatorial markets. In a general setting of combinatorial markets with item pricing, we have many buyers, each having a publicly known valuation function for each subset of purchased items, and many strategic sellers, each having a set of items to sell at a price for item . The utility of a buyer for subset is where . In general, the valuation function for a buyer can be quite complicated. The strategy of each seller is to set prices to maximize the prices for the items they sell. When items have a cost to produce, the seller wants to maximize their prices minus the costs for items sold. In this generality, the existence and properties (e.g., uniqueness, market clearing or not) of possible pure Nash Equilibria (NE) will depend on the precise setting.
The main focus of the Babaioff et al paper is for a single buyer and strategic sellers each having a single copy of a distinct item. They establish basic results for different classes of valuation functions. Namely, they consider (in order of decreasing generality), arbitrary, subadditive, XOS, submodular, and gross substitutes valuations^{1}^{1}1A subadditive valuation is such that . A XOS valuation is such that for . A gross substitutes valuation is such that if at a price p, a set is purchased and at a price , a set is purchased then . We will not consider these valuation classes but include these definitions for completeness.. In our setting, we will restrict attention to submodular valuations and additive valuations. Babaioff et al also discuss multiple buyers in a Bayesian setting and how their results can be extended in the case of costs. In a following paper, Lev, Oren and Boutilier [6] consider the setting of a single monotone submodular buyer where now each seller has a number of different items for sale. Our contribution will be to further the work of Babaioff et al with regard to more than one submodular buyer in a full information setting. Our work generalizes the previous settings in that we have multiple buyers each with their own valuations. We also assume that sellers have costs to produce an item. We extend the setting of Lev et al to multiple buyers, but restrict to each seller only selling copies of a distinct item. Hence, overall, our setting is incomparable to theirs.
2 Related results
Walrus [7] pioneered the study of markets and market equilibrium (for divisible items being traded) in the late 19th century. Market equilibria is often referred to as competitive equilibria or Walraisian equlibria. The adaption of markets to buyers (having money) and sellers (with divisible items) is called Fisher markets (attributed to Irving Fisher and also dating back to the late 19th century; see Brainard and Scarf [4]). With regard to indivisible items, Gul and Stracchetti [5] provide a fundamental result showing that the class of Gross Substitutes valuations (a strict subset of submodular valuations) is the largest class of valuations containing unit demand buyers that always have a Walraisian equilibrium. Amongst other conditions, such equilibria require that prices are set so that all items are sold (i.e., the prices are a market clearing Nash equilibrum).
The topic of pricing is too extensive to indicate all the related work. It is rather remarkable, however, that precise characterization of pricing equilibria is a relatively recent topic of interest. As already indicated , our work is most closely related to the comprehensive paper of Babaioff et al [2] for a single buyer and the following extension by Lev et al [6] to sellers who can have more than one item for sale. Borodin, Lev and Strangway [3] consider another extension of Babaioff et al. Namely, they return to a single item per seller and a single buyer setting but now impose a budget on the buyer. We will not consider budgets although clearly budgets are an important consideration. Budgets raise many additional considerations for buyers and the resulting seller prices. This was already evident in the single buyer case. In our appendix, we will just illustrate how even in a very restricted setting, budgets will complicate the analysis of possible equilibria for multiple additive buyers.
3 Preliminaries and Summary of Results
We assume that is the set of sellers. Here, we use to denote the set . We initially assume that seller sells item and has copies of this item to possibly sell to different buyers. We also assume there is a cost to seller to produce each copy of their item. Each seller sets a price which is same for all copies of item .
We assume that is the set of buyers and each buyer has a submodular valuation function where is the set of all nonnegative real numbers. We assume that each buyer is interested in buying at most one copy of each item. A submodular function is a function such that , and . We also assume that is normalized () and monotone ( such that ). For a given pricing , the utility of a buyer for a subset of items is given by where . Each buyer chooses a set such that . There can be more than one set that maximizes the utility, in which case a buyer arbitrarily (but deterministically) chooses a set that has the highest cardinality. Thus every pricing p induces a given allocation of items to each buyer . The social welfare of an allocation induced by a pricing is defined as the sum .
There are different pricing games that can be considered when there are many buyers and sellers. In the mechanism design area, we usually study combinatorial auctions for a single seller and multiple buyers who are the strategic agents. In contrast, in the pricing game as initiated in Babaioff et al, there is one utility maximizing buyer and it is the sellers who are the strategic agents whose actions consist solely of setting prices. But once there is a limited supply (i.e., more buyers than copies) of each item, it is not clear which buyers will obtain a given item. Since we do not view the buyers as being strategic (i.e., their valuations are true and known), either the sellers or an outside mechanism have to determine an order in which to serve buyers. That is, we are essentially considering an online algorithm in which buyers arrive sequentially. In the limited supply case, we will study such an online setting and leave other extensions for future work.
A pure Nash Equilibrium (PNE) of the game is given by a pricing . Under this pricing p, no seller can unilaterally increase profit by changing their price. This is an equilibrium in the game between sellers. The profit of a seller is given by where denotes the number of buyers that buy item and is the cost of production to seller . The cost of production to one or more sellers may be 0. A market clearing Pure Nash Equilibrium is a PNE such that all copies of all the items are sold by the sellers. In addition to considering the standard Nash equilibrium, we will sometimes consider Nash equilibria for sufficiently small in which any individual seller cannot gain more than an additive profit of by deviating.
Let . The price of anarchy (POA) for a market game instance is then defined as . That is, the POA is the worst case approximation induced by a PNE. The price of stability (POS) is defined as . That is, the best case approximation induced by a PNE.
As stated, throughout this paper we assume that the buyer valuation functions are monotone submodular for each . For some results, we further assume that the are additive for one or more buyers. We define as for , , . We denote by the marginal value . We note that , and by the definition of submodularity.
3.1 Summary of results
In sections 4 and 5 (and also Appendix A), we have as many copies of items as we have buyers. Hence, buyers are never in competition for an item. In sections 4 and 5 we provide a rather comprehensive set of results including a characterization for when there is a unique market clearing pure equilibirium and a sufficient condition for when there is a unique equilibrium. In the unlimited supply setting with no budgets, we consider the social welfare obtained in equilibria, and the resulting price of anarchy and price of stability. Appendix A considers a very restricted setting in which a budget is introduced. In this setting, we do not have explicit prices but rather we have a set of conditions that can easily be tested and these conditions are both necessary and sufficient for a market clearing equilibrium.
In section 6, we have a limited supply of items. As stated, we need to have some means to resolve conflicts amongst the buyers. In particular, we will assume that each seller has only one copy of their item. We begin with some general observations that hold no matter how items are allocated to buyers. In section 6.1, we consider the restricted setting where there is one submodular buyer and the remaining buyers are additive. We assume that the sellers (or a central authority) have agreed upon a fixed order in which agents arrive and get their choice of items. The negative results (i.e., Examples 3.1, 6.1) therefore hold in the online setting where an adversary sets the order of arrivals. This is similar to online posted price mechanisms as studied for the case of a single seller. Here, however, we have more than one seller and for any positive claims about equilibria, we will assume that buyers arrive in a fixed order that is known to all sellers.
Our positive results for the limited supply setting as with regard to the existence and characterization of pure Nash equilibira and market clearing equilibria are restricted to some very special settings. Our negative examples suggest that any extension of our positive results to more general markets (with multiple submodular buyers) will not follow in any immediate way.
As an indication of the difference between previous results for a single buyer discussed in [2] where there is always a pure Nash equilibrium in the case of submodular valuations, we provide an example where no Nash Equilibirum exists in the limited supply setting. In the following example there is no Nash equilibrium for a sufficiently small .
Consider the market where , and the valuation function is given by the following table:
sets  1  2  1, 2 

16  16  32  
30  30  45 
is additive and is submodular as can be easily verified. There is no pure Nash equilibrium for when buyer is preferred by every seller over buyer at a given pricing when both the buyers are willing to purchase. At any given pricing, buyer chooses a set that it consumes, and then buyer chooses a subset of that maximizes its utility.
Proof.
Let and suppose there is an Nash Equilibrium p and under pricing p, is sold to and to . The equilibrium has to be market clearing because each seller can earn a profit of at least by selling to buyer 1 at a price of 16.
Let and . Then, since otherwise buyer 1 will not purchase those items. Now, if seller increases the price to 30, buyer 1 will purchase set and buyer 2 will purchase increasing the profit of seller 2 by more than . So, there is no equilibrium with , .
Let and . Then, since otherwise buyer 1 will not purchase item and since buyer 1 does not purchase but buyer 2 does. If , seller can increase its price to 23 and be sold to buyer 2 increasing the profit by more than . If , seller can increase the price to 30 and still be sold to buyer 2 increasing the profit by more than . A similar thing happens for the case of , . So, no equilibria exists in these cases.
Let and . Then. since otherwise the buyer 2 will purchase at most one item. At this price, buyer 1 will already purchase both the items and hence this case is not possible.
Thus, an Nash equilibrium does not exist in this case. ∎
4 Characterization of Market Clearing Equilibrium in case of two copies and two submodular buyers
We begin by considering the case of two buyers and two copies of each item, i.e., . In this case, the buyer order does not matter. Also, here market clearing means that each buyer buys from each seller. We establish the necessary and sufficient conditions for the existence of market clearing pure Nash equilibrium as stated in the following theorem, and proved using observation 4, lemma 4 and lemma 4. The proof of this theorem along with other proofs in this section are subsumed in section 5 but for notational convenience, we state them separately.
The necessary and sufficient condition in the case of 2 buyers () and n sellers () for the existence of a market clearing pure Nash equilibrium is that and . Moreover, when this condition is satisfied, the market clearing pure Nash equilibrium is unique.
We contrast this theorem by giving an example where the condition is not satisfied and no Nash equilibrium exists.
Consider the market where , , the cost to sellers is and the valuation function is given by the following table:
sets  1  2  1, 2 

200  210  220  
70  50  120 
Valuation is submodular and is additive as can be easily verified. We will show that for , there cannot be a pure Nash equilibrium for these valuations.
Proof.
Let p be an Nash equilibrium for some . Let be the set purchased by buyer 1 and be the set purchased by buyer 2 at this pricing. There are four possibilities for each and namely and . We will consider all possibilities and in each case show that some seller can improve upon their revenue by deviating. We do not need to consider the combinations where since every seller can sell to buyer 2 by setting a price earning strictly more profit than and hence .

If , then since otherwise buyer 1 will prefer over and hence seller 1 can profit by more than by setting a price of and just selling to buyer 2.

If and , then and since otherwise buyer 1 will prefer over . If then by setting a price of , seller 2 will earn a profit of increasing it by more than . If , seller 1 can profit by more than by setting a price of and just selling to buyer 2.

The case of and is not a possible equilibrium since in this case and hence buyer 1 will prefer over .

If and , then , and since otherwise buyer 1 will prefer over . If then by setting a price of , seller 1 will increase his profit by more than . If and , seller 2 can profit by more than by setting a price of . If and , then by setting a price of seller 1 can sell to both buyers earning a profit of increasing it by at least .

The case of and is not possible since in this case and , so buyer 2 will purchase .

If and , then and . By setting a price of 40, seller 1 can sell to both the buyers and earn a profit of increasing the profit by at least .

If and , then and . This case is not possible since at this pricing, buyer 1 will prefer over .
So, p is not an Nash equilibrium for . Since p is arbitrary, there is no Nash equilibrium for . ∎
We first prove that the given condition is a necessary condition. We start by making three observations that will be helpful in the proof.
Observation .
For a buyer with submodular valuation function , if some item has a price of at most , the buyer will prefer to purchase item .
Proof.
Let us consider a pricing p such that . Let us assume that the buyer will buy a set of items that does not contain under this pricing p. Then, (since is submodular) and hence , so under pricing p, the buyer will prefer to purchase due to the tiebreaking rule which breaks ties in favor of larger sets. ∎
Observation .
For a buyer with submodular valuation function , if some item has a price of at most for some , , the buyer will prefer to purchase over .
Proof.
Let us consider a pricing p such that . Then, and hence , so under pricing p, the buyer will prefer to purchase over due to the tiebreaking rule which breaks ties in favor of larger sets. ∎
Observation .
Let , and suppose there exists such that . Then, there is no market clearing pure Nash equilibrium.
Proof.
Let us assume, for the sake of contradiction, that there is a market clearing pure Nash equilibrium given by pricing p. since if , then since and hence buyer 2 will purchase instead of . Similarly, . Thus, profit of seller is and hence seller will earn more profit by deviating to price . Since seller has an incentive to deviate, p is not an equilibrium.
Since p was assumed to be an arbitrary market clearing pure Nash equilibrium, there is none. ∎
Let , and suppose there exists such that . Then, there is no market clearing pure Nash equilibrium.
Proof.
Let us assume, for the sake of contradiction, that there is a market clearing pure Nash equilibrium given by pricing p. since if , then since and hence buyer 2 will purchase instead of .
Since p is market clearing, both buyers will purchase all items and hence the profit of seller will be . If seller instead sets price , buyer will still purchase item due to observation 4 and new profit of seller will be . So, seller has an incentive to deviate from p and hence p is not an equilibrium.
Since p was assumed to be an arbitrary market clearing pure Nash equilibrium, there is none. ∎
Lemma 4 proves that is a necessary condition for the existence of a market clearing pure Nash equilibrium in the case of 2 buyers. Similarly, is also a necessary condition. Thus, is a necessary condition for the existence of a market clearing pure Nash equilibrium in the case of 2 buyers. Also, from observation 4, is a necessary condition. Now, we will prove that these two conditions together are also a sufficient condition.
There is a unique market clearing pure Nash equilibrium p in the case of when and . Namely, .
Proof.
Let .
Since and , it is clear that both the buyers will purchase all the items due to observation 4. Thus, there is no profit in reducing the price for any seller. The profit in this case for seller is .
Without loss of generality, we consider the case where for some . In this case, . Let us assume that seller increases the price of his item to and this new pricing is called . Let us also assume that buyer purchases a set containing under this new pricing. If , then for all , , so since . Hence buyer will prefer over due to tiebreaking rule. This is a contradiction and hence . Since , , so buyer 2 will not purchase . Similarly if seller increases the price to more than , buyer will not purchase item . So, the maximum profit that seller can earn by increasing the price is . Since , there is no incentive to increase the price. Therefore, p is a pure Nash equilibrium.
Let us assume that there is another market clearing pure Nash Equilibrium such that for some . If , then buyer will prefer to buy set over set since and hence . Therefore, . Similarly, . If , then by setting price , seller will still sell both the copies of his item due to observation 4 and hence will earn strictly more profit. Thus, is not a market clearing pure Nash Equilibrium if for some .
Hence, p is the unique market clearing pure Nash equilibrium. ∎
When for some , there are no market clearing pure Nash equilibria as stated in Observation 4. Market clearing pure Nash equilibria are also total welfare maximizing in the case that since total welfare is the sum of the utility of all buyers and sellers. When buyer 1 buys a set and buyer 2 buys a set , the money is transferred from buyers to sellers and hence the total welfare is which is maximum when , that is when the market is cleared. Thus, when market clearing pure Nash equilibria exists, the price of stability is 1.
For the case of a single submodular buyer, and each seller having a single copy of their distinct item, Babaioff et al [2] show that there is a unique pure Nash equilibrium (NE) which is market clearing and that no other pure NE can exist. That is, the one and only pure Nash is market clearing. We now show that in the setting of two buyers and each seller having two copies of their item, there can be other Nash equilibrium that are not market clearing. These nonmarket clearing NE can exist even when the necessary and sufficient conditions hold for the unique market clearing NE. Consider the case of two additive buyers, where for some seller , the seller is indifferent between setting price and since in both cases, the seller will earn equal profit. So, the pure Nash equilibrium in which it sets a price of is not market clearing since buyer 2 will not purchase at this price. So, in case or for some , there are pure Nash Equilibria which are not market clearing. Also in the case of two additive buyers, if , seller will always earn nonpositive profit and hence can set any arbitrarily high price, and raising or lowering that price does not increase the profit of seller . So in this case as well, there can be many pure Nash Equilibria which are not market clearing. Furthermore, even if the inequalities in the characterization of market clearing equilibria are strict inequalities, there can still be pure Nash equilibria that are not market clearing. That is, even when and , the uniqueness of pure Nash equilibrium is not guaranteed. In particular, there can sometimes be pure Nash Equilibrium that are nonmarket clearing when both the buyers are submodular.
Consider the market where , , the cost to sellers is and the valuation function is given by the following table:
sets  1  2  3  1, 2  1, 3  2, 3  1, 2, 3 

10  10  10  14  15  13  16  
10  10  10  13  15  14  16 
Both the valuation functions are submodular as can be easily verified. The pricing with and is a pure Nash equilibrium which is not market clearing.
In contrast to the previous example, we can show the uniqueness of pure Nash equilibrium when at least one of the valuation functions is additive. Without loss of generality, we assume that is submodular and is additive.
When , , is submodular, is additive, and and then there is only one pure Nash Equilibrium and that equilibrium is market clearing. This equilibrium is given by .
Proof.
Let p be an arbitrary pure Nash equilibrium. For each , since if is smaller than both and , seller can increase the price and still sell to both the buyers as proved earlier in observation 4. This will only increase his profit.
Let and be the sets purchased by the two buyers respectively. Let be arbitrary. Then, since otherwise buyer 1 will prefer over due to observation 4. Also, since is additive and hence buyer 2 will purchase at maximum price . Since , it is better for seller to set the price equal to since at this price, due to submodularity and hence in this case both the buyers will purchase item due to the tiebreaking rule and seller will earn more profit. Thus, the set is empty.
If for some seller , , then by setting price , the seller will sell to both buyers and earn a positive profit. So, in equilibrium. Therefore, .
Let some . In this case, since otherwise seller can earn more profit by increasing the price. Also, since otherwise is more valuable to buyer 1 than . Therefore, . In this case, and setting price is more profitable since by doing so, the profit increases to . Thus, the set is empty.
Therefore, . Therefore, and since otherwise at least one of the buyers will not purchase . Hence, gives a unique pure Nash Equilibrium and this equilibrium is market clearing. ∎
Note that the price can be as high as in case is an arbitrary submodular valuation and it might hold that in which case selling to both buyers is not profitable for seller and hence it is not necessary that is empty. Thus, the proof will not follow.
The following corollary follows from the above theorem.
When , , is submodular, is additive, and and , the price of anarchy is 1.
5 The Unlimted Supply Setting for an Arbitrary Number of Buyers
Generalising the case of two buyers, the necessary and sufficient condition in case of buyers and copies per seller (i.e., ) is given by the following theorem. We prove it using observation 5, lemma 5 and lemma 5. Recall that .
For all , we define to be a permutation of in nondecreasing order of ( and hence ).
The necessary and sufficient condition in case of m buyers () and n sellers () for the existence of a market clearing pure Nash equilibrium is and .
The necessary condition follows from the following observation and lemma.
Observation .
Let , and suppose there exists such that . Then, there is no market clearing pure Nash equilibrium.
Proof.
Let us assume, for the sake of contradiction, that there is a market clearing pure Nash equilibrium given by pricing p. since if , then since and hence buyer will purchase instead of . Thus, . Thus, profit of seller is and hence seller will earn more profit by deviating to price . Since seller has an incentive to deviate, p is not an equilibrium.
Since p was assumed to be an arbitrary market clearing pure Nash equilibrium, there is none. ∎
If for some and some , , then there is no market clearing pure Nash Equilibrium.
Proof.
Let us assume to the contrary that there is a market clearing pure Nash Equilibrium p. Then, since otherwise some buyer will not purchase item since the utility from will be strictly higher than from . Also, since otherwise seller can increase price and still sell to all the buyers as shown in observation 4. Therefore, . Thus, the profit of seller under this pricing is .
Now, if seller deviates to a price of , it will still sell to buyers since new price . Thus, the new profit of seller is and hence the seller has an incentive to deviate. Therefore, p is not a market clearing pure Nash Equilibrium.
Since p was assumed to be an arbitrary market clearing pure Nash Equilibrium, there is none. ∎
The sufficient condition follows from the following lemma.
If and , then there is a unique market clearing pure Nash Equilibrium.
Proof.
Let p be a pricing such that .
At this pricing, all the items are sold to all the buyers as shown in observation 4 since . Thus, no seller has an incentive to lower the price. The profit to seller at this pricing is .
Let us consider that some seller increases its price to and let the new pricing be . Since , every buyer will purchase all the items other than . Let be the minimum index such that . If there is no such , then no buyer will purchase item and hence this deviation is not profitable to seller . Item will be purchased by sellers and hence the maximum profit seller will earn is and hence it is not profitable for seller to deviate.
Thus, p is a market clearing pure Nash Equilibrium.
Let us assume that there is another market clearing pure Nash Equilibrium . Since it is market clearing, . If for some , then deviating to price is profitable to seller since it will still sell to all the buyers and hence seller will earn strictly more profit at this new price.
Thus, p is the unique market clearing pure Nash Equilibrium. ∎
We now show that theorem 4 extends to the case of m buyers when at most one of them is submodular.
When , , the valuation of one of the buyer is submodular and the valuations for the remaining buyers are additive, and and , then there is only one pure Nash Equilibrium and that equilibrium is market clearing. This equilibrium is given by .
Proof.
Let p be an arbitrary pure Nash equilibrium. For each , since if is smaller than , seller can increase the price and still sell to all the buyers as proved earlier in observation 4. This will only increase his profit.
Let there be an item that is not purchased by the submodular buyer at pricing p. Let be the submodular buyer for some . Then, for some and item is purchased by buyers (a total of m  k + 1) buyers. Thus, the profit of seller is which will be the profit if the seller sets a price of . Thus, p is not an equilibrium. This is a contradiction and hence every item is purchased by the submodular buyer in an equilibrium. Therefore, in an equilibrium, the price of item is at most where is the submodular buyer.
Let us assume that for some . Then, for some since . Now if the seller increases the price to , the set of buyers that purchase item does not change and hence the profit of increases. Therefore, this is not possible in an equilibrium. Thus, for some for all .
Let for some for some . Then, the buyers purchases the item and hence the profit of seller is . The seller can increase his profit by setting a price of since . Thus, this is not possible in an equilibrium.
Therefore, gives the unique pure Nash equilibrium. This equilibrium is market clearing since all the buyers purchases all the items. ∎
The following corollary follows from the above theorem.
When , , the valuation of one of the buyer is submodular and the valuations for the remaining buyers are additive, and and , the price of anarchy is 1.
Next we give an example in case of buyers in which we show that the price of anarchy can be as high as .
Consider the market where , , , for some and . In this case, there is an equilibrium in which the only seller sets a price of and at this price, only buyer purchases. The welfare of this equilibrium is . The optimal welfare, however, is (where is harmonic number, which is approximately ) which is achieved when all the buyers purchases the item (at a price at most ). Therefore, for this equilibrium, the welfare is at most of the optimal.
Note that the example above can be extended to the case of sellers by taking additive valuations
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