Limits of Efficiency in Sequential Auctions

Limits of Efficiency in Sequential Auctions

Abstract

We study the efficiency of sequential first-price item auctions at (subgame perfect) equilibrium. This auction format has recently attracted much attention, with previous work establishing positive results for unit-demand valuations and negative results for submodular valuations. This leaves a large gap in our understanding between these valuation classes. In this work we resolve this gap on the negative side. In particular, we show that even in the very restricted case in which each bidder has either an additive valuation or a unit-demand valuation, there exist instances in which the inefficiency at equilibrium grows linearly with the minimum of the number of items and the number of bidders. Moreover, these inefficient equilibria persist even under iterated elimination of weakly dominated strategies. Our main result implies linear inefficiency for many natural settings, including auctions with gross substitute valuations, capacitated valuations, budget-additive valuations, and additive valuations with hard budget constraints on the payments. Another implication is that the inefficiency in sequential auctions is driven by the maximum number of items contained in any player’s optimal set, and this is tight. For capacitated valuations, our results imply a lower bound that equals the maximum capacity of any bidder, which is tight following the upper-bound technique established by Paes Leme et al. [19].

1 Introduction

Consider the following natural auction setting. An auction house has a number of items that are offered for sale in an auction on a particular day. To orchestrate this, the auction house publishes a list of the items to be sold and the order in which they will be auctioned off. The items are then sold one at a time in the given order. A group of bidders attends this session of auctions, with each bidder being allowed to participate in any or all of the single-item auctions that will be run throughout the day. Since the auctions are run one at a time, in sequence, this format is referred to as a sequential auction.

This way of auctioning multiple items is prevalent in practice, due to its relative simplicity and transparency. It also arises naturally in electronic markets, such as eBay, due to the asynchronous nature of the multiple single-item auctions that are executed on the platform. A natural question, then, is how well such a sequential auction performs in practice. Note that while the auction of a single item is relatively simple, equilibria of the larger game may be significantly more complex. For instance, a bidder who views two of the items as substitutes might prefer to win whichever sells at the lower price, and hence when bidding on the first item he must look ahead to the anticipated outcome of the second auction. What’s more, the sequential nature of the mechanism implies that the outcome of one auction can influence the behavior of bidders in subsequent auctions. This gives rise to complex reasoning about the value of individual outcomes, with the potential to undermine the efficiency of the overall auction.

In this work we study the efficiency of sequential single-item first-price auctions, where items are sold sequentially using some predefined order and each item is sold by means of a first-price auction. We study the efficiency of outcomes at subgame perfect equilibrium, which is the natural solution concept for a dynamic, sequential game. Theoretical properties of these sequential auctions have been long studied in the economics literature starting from the seminal work of Weber [23]. However, most of the prior literature has focused on very restricted settings, such as unit-demand valuations, identical items, and symmetrically distributed player valuations. The few exceptions that have attempted to study equilibria when bidders have more complex valuations tend to have other restrictions, such as a very limited number of players or items [11, 20, 3, 2]. Much of the difficulty in studying these auctions under complex environments and/or valuations stems from the inherent complexity of the equilibrium structure, which (as alluded to above) can involve complex reasoning about future auction outcomes.

Paes Leme et al. [19] and Syrgkanis and Tardos [21] circumvented this difficulty by performing an indirect analysis on efficiency using the price-of-anarchy framework. They showed that when bidders have unit-demand valuations (UD), items are heterogeneous, and bidders’ valuations are arbitrarily asymmetrically distributed, then the social welfare at every equilibrium is a constant fraction of the optimal welfare. Syrgkanis and Tardos [22] extended this result to no-regret learning outcomes and to settings with budget constraints. On the negative side, Paes Leme et al. [19] showed that this result does not extend to submodular valuations (SM): there exists an instance with submodular valuations where the unique “natural” subgame perfect equilibrium leads to inefficiency that increases linearly with the number of items, even for a constant number of bidders.

The above results leave a large gap between the positive regime (unit-demand bidders) and the negative (submodular bidders). Many natural and heavily-studied classes of valuations fall in the range between UD and SM valuations. Among them are the following, arranged roughly from most to least general:

  • Gross-substitutes valuations (GS): A valuation satisfies the gross-substitutes valuation property if, whenever the cost of one item increases, this cannot reduce the demand for another item whose price did not increase.

  • -capacitated valuations (-CAP): Each player has a capacity and a value for each item; the value for a set of items is then the value of the highest-valued items in the set.

  • Budget-additive valuations (BA): The value of a player is additive up to a player-specific budget and then remains constant.

The class of GS valuations is motivated by the fact that it is (in a certain sense) the largest class of valuations for which a Walrasian equilibrium is guaranteed to exist [12], and a Walrasian equilibrium, if exists, is always efficient (see, e.g., [6]). It is known that every -capacitated valuation satisfies gross substitutes [9]. Moreover, every gross substitutes valuation is submodular [15], and it is easy to see that unit-demand valuations are precisely -capacitated valuations. We therefore have UD -CAP GS SM. The set of budget-additive valuations is incomparable to UD, -CAP, and GS, but it is known that BA SM.

We ask: for which of the above classes does the sequential first-price auction obtain a constant fraction of the optimal social welfare at equilibrium? In this work we show that the answer to the above question is none of them.

Specifically, we show that for the case of gross substitutes valuations and for budget additive valuations, the inefficiency of equilibrium can grow linearly with the number of items and the number of players. Thus, even for settings in which a Walrasian equilibrium is guaranteed to exist, an auction that handles items sequentially cannot find an approximately optimal outcome at equilibrium. For the case of -capacitated valuations, we show that the inefficiency can be as high as . This bound of is tight, following the upper bound established by [19].

To prove these lower bounds we consider a different, conceptually more restrictive, class of valuations: the union of unit-demand and additive valuations. We construct an instance in which every bidder has either a unit-demand valuation or an additive valuation, then show that the unique “natural” equilibrium for this instance has extremely poor social efficiency. We then adapt this construction to provide a lower bound for the valuation classes described above.

We also extend our lower bound to apply to one other setting: additive valuations when players have hard budget-constraints on their payments. This setting falls outside the quasi-linear regime, but is very relevant in the sequential auction setting: for instance, each bidder may arrive at an auction session with only a certain fixed amount of money to spend. Note that this is different from the BA valuation class, since it does not restrict the value of a player for a set of items, but rather limits the total payment that a player can make. For this setting, it is known that maximizing welfare is not an achievable goal in most auction settings, as a participant with low budget is necessarily ineffective at maximizing the value of the item(s) she obtains. Instead, the natural notion of social efficiency is the “effective welfare,” in which the contribution of each participant to the welfare is capped by her budget [22]. We show that, even comparing against the benchmark of effective welfare, our negative result also applies to this setting: for additive valuations with hard budget constraints, the inefficiency can grow linearly with the number of items or players. This is in stark contrast to the setting of simultaneous first-price auctions, where it is known that a constant fraction of the optimal effective social welfare occurs at equilibrium for bidders with hard budget constraints, even when valuations are fractionally subadditive [22] (where this class falls between submodular and subadditive valuations).

Sequential auctions with additive bidders and hard budget constraints have been studied in only very limited settings in the economics literature and have recently begun to attract the attention of the computer science community [14]. Our result shows that if one allows for arbitrary additive valuations, then such an auction process can lead to very high inefficiency.

All of the negative results described above rely heavily on the fact that items can be sold in an arbitrary order. This leads naturally to the following design question: does there always exists an order on the items that results in better outcomes at a subgame perfect equilibrium? This can be interpreted as a mechanism design problem, in which the auctioneer wishes to choose the order in which items are sold in order to mitigate the social impact of strategic bidding. We conjecture that a concrete class of item orders (that we propose) always contains a good order that leads to the VCG outcome at equilibrium, for the class of single-valued unit-demand valuations. We leave the resolution of this conjecture as an open problem.

1.1 Related Work

Sequential auctions have been long studied in the economics literature. Weber [23] and Milgrom and Weber [17] analyzed first- and second-price sequential auctions with identical items and unit-demand bidders in an incomplete-information setting and showed that the unique symmetric equilibrium is efficient and the prices have an upward drift. The behavior of prices in sequential studies was subsequently studied in [1, 16]. Boutilier el al. [7] studies first-price auctions in a setting with uncertainty, and devised a dynamic-programming algorithm for finding the optimal strategies (assuming stationary distribution of others’ bids).

The setting of multi-unit demand has also been studied under the complete-information model. Several papers studied the two-bidder case, where there is a unique subgame perfect equilibrium that survives the iterated elimination of weakly dominated strategies (IEWDS) [11, 20]. Bae et al. [3, 2] studied the case of sequential second-price auctions of identical items with two bidders with concave valuations and showed that the unique outcome that survives IEWDS achieves a social welfare at least of the optimum. Here we consider more than two bidders and heterogeneous items.

Recently, Paes Leme et al. [19] analyzed sequential first- and second-price auctions for heterogeneous items and multi-unit demand valuations in the complete-information setting. For sequential first-price auctions they showed that when bidders are unit-demand, every subgame perfect equilibrium achieves at least 1/2 of the optimal welfare, while for submodular bidders the inefficiency can grow with the number of items, even with a constant number of bidders. The positive results were later extended to the incomplete-information setting in [21] and to no-regret outcomes and budget-constrained bidders in [22]. In this work we close the gap between positive and negative results and show that inefficiency can grow linearly with the minimum of the number of items and bidders even when bidders are either additive or unit-demand.

This work can be seen as part of the recent interest line of research on simple auctions. The closest literature to our work is the that of simultaneous item-bidding auctions [5, 8, 4, 13, 10, 22], which is the simultaneous counterpart of sequential auction. In contrast to sequential auctions, in simultaneous item auctions constant efficiency guarantees have been established for general complement-free valuations, even under incomplete-information settings or outcomes that emerge from learning behavior. We refer to [18] for a recent survey on the efficiency of simultaneous and sequential item-auctions.

2 Model and Preliminaries

We consider settings with bidders and items, where every bidder has a valuation function , associating a non-negative real value with every subset of items. We denote the set of bidders by and the set of items by . The valuation function is assumed to be monotone (i.e., for every ). An allocation is a vector , where denotes the set of items allocated to bidder , and such that for every .

Sequential item auctions.

The auction proceeds in steps, where a single item is sold in every step using a first-price auction. In every step , every bidder offers a bid , and the item is allocated to the agent with the highest bid for a payment that equals his bid. Each bid in each step can be a function of the history of the game, which is assumed to be visible to all bidders. More formally, a strategy of bidder is a function that, for every step , associates a bid as a function of the sequence of the bidding profiles in all periods . The utility of an agent is defined, as standard, to be his value for the items he won minus the total payment he made throughout the auction (i.e., quasi-linear utility). We will also assume that the bid space is discretized in small negligible -increments, and for ease of presentation we will use to denote the bid .

This setting is captured by the framework of extensive-form games (see, e.g., [19]), where the natural solution concept is that of a subgame-perfect equilibrium (SPE). In an SPE, the bidding strategy profiles of the players constitute a Nash equilibrium in every subgame. That is, at every step and for every possible partial bidding profile up to (but not including) step , the strategy profile in the subgame that begins in step constitutes a Nash equilibrium in the induced (i.e., remaining) game.

Elimination of Weakly Dominated Strategies.

We wish to further restrict our attention to “natural” equilibria, that exclude (for example) dominated overbidding strategies. We therefore consider a natural and well-studied refinement of the set of subgame perfect equilibria: those that survive iterated elimination of weakly dominated strategies (IEWDS). A strategy is weakly dominated by a strategy if, for every profile of other players’ strategies , we have , and moreover there exists some such that . Roughly speaking, under IEWDS, each player removes from her strategy space the set of all weakly dominated strategies. This removal may cause new strategies to become weakly dominated for a player, which are then removed from her strategy space, and so on until no weakly dominated strategies remain. We defer a formal definition of IEWDS to Appendix A.

We will focus on subgame perfect equilibria of sequential first-price item auctions that survive IEWDS. It is shown in [19] that there always exists such an equilibrium. We note one necessary property of an equilibrium satisfying IEWDS: in every subgame beginning at a time (i.e., when the last item is being sold), for every possible bidding history up to that round, each player will bid no more than his marginal value for the final item. In other words, no player can credibly threaten to overbid on the last item for sale.

Price of anarchy.

The price of anarchy (PoA) measures the inefficiency that can arise in strategic settings. The PoA for subgame perfect equilibria is defined as the worst (i.e., largest) possible ratio between the welfare obtained in the optimal allocation and the welfare obtained in any subgame perfect equilibrium of the game. We note that all of our lower bounds on the price of anarchy will involve “natural” equilibria that survive IEWDS.

3 A Simple Example

To develop some intuition regarding the strategic considerations that might take place in sequential auctions, we give a simple example in which one bidder has value for many items (i.e., wholesale buyer) and another bidder has value for only one item (i.e., retail buyer).

In particular, consider a sequence of two auctions for two identical items and two buyers, and . Buyer is a “wholesale” buyer, having an additive valuation with a value of for each of the two items. Buyer is a “retail” buyer, who wants only one item (unit-demand) and has a value of for either of the two. The items are sold sequentially using a first-price auction for each item.

Consider the situation from the perspective of the additive buyer . Thinking strategically and farsightedly, he reasons that if he wins the first auction, then in the second auction he will have to compete with buyer and will therefore have to pay dollars to win the second item. If, however, he lets buyer win the first item, then buyer will have no value for the second item and hence the only undominated strategy for buyer will be to bid in the second auction, and hence buyer will win the second item for free. What must buyer pay in order to win the first item? Buyer knows that if the first item goes to buyer , then buyer will certainly lose the second item as well; therefore buyer is willing to pay up to for the first item. Therefore, in order to win the first item, buyer will have to bid at least in the first auction.

Thus bidder needs to choose between the following two options: he can either win both auctions and pay a price of for each one of them, or let bidder win the first auction and win only the second auction but pay nothing. Observe that the first option gives bidder a utility of () while the second option gives him a utility of (). Consequently, bidder will choose to forego the first item in order to improve his situation in the second one. Interestingly, this outcome is socially suboptimal, since the efficient outcome is for bidder to win both items — although bidder has much more value for the first item than bidder , the first item is allocated to in equilibrium.

One can also take this example to the extreme where, e.g., bidder ’s value is set to for each item. In this case the unique subgame perfect equilibrium that survives elimination of dominated strategies is a approximation to the optimal welfare, even though the items are identical (and therefore the inefficiency is irrespective of item ordering). In the next section we demonstrate that with heterogeneous items, the social welfare of sequential item auctions at subgame perfect equilibrium can be as low as an fraction of the optimal social welfare.

4 Lower Bound for Additive and Unit-demand Valuations

We now present our main result by providing an instance of a sequential first price auction with unit-demand and additive bidders, where the social welfare at a subgame-perfect equilibrium that survives IEWDS1 achieves social welfare that is only an -fraction of the optimal welfare. Therefore, our example shows that inefficiency can arise at equilibrium in a robust manner.

Theorem 4.1

The price of anarchy of the sequential first-price item auctions with additive and unit-demand bidders is . Moreover, this result persists even if we consider only equilibria that survive IEWDS.

Informal Description. Before we delve into the details of the proof of Theorem 4.1, we give a high-level idea of the type of strategic manipulations that lead to inefficiency and compare them with the simultaneous auction counterpart of our sequential auction.

Consider an auction instance where two additive bidders have identical values for most of the items for sale, but their valuations differ only on the last few items that are sold. Specifically, assume that there are two items and , auctioned last, such that only player has value for and only player has value for . We will refer to these items as the non-competitive items and to all other items as the competitive items. The additive bidders know that it is hopeless to try to achieve any positive utility from the competitive items on which they have identical interests. The only utility they can ever derive is from the last, non-competitive items on which they don’t compete with each other. If these were the only two players in the auction, then we would obtain the optimal outcome: the two bidders would simply compete on each of the competitive items, with one of them acquiring each competitive item at zero utility.2

We now imagine adding unit-demand bidders to the auction in order to perturb the optimality. Specifically, suppose there is a unit-demand bidder that has value for the two non-competitive items, with the value for item being slightly less than player ’s value for , . This endangers the additive bidders’ hopes of getting non-negligible utility, since competition from the unit-demand player may drive up the prices of and . The only hope that the additive bidders have is that the unit-demand bidder will have his demand satisfied prior to these final two auctions, in which case the unit-demand bidder would not bother to bid on them. Hence, the two additive bidders would do anything in their power to guide the auction to such an outcome, even if that means sacrificing all the competitive items! This is exactly the effect that we achieve in our construction. Specifically, we create an instance where this competing unit-demand bidder has his demand satisfied prior to the auctions for and if and only if a very specific outcome occurs: the additive bidders don’t bid at all on all the competitive items, but rather other small-valued bidders acquire the competitive items instead. These small-valued bidders contribute almost nothing to the welfare, and therefore all of the welfare from the competitive items is lost.

It is useful to compare this example with what would happen if the auctions were run simultaneously, rather than sequentially. This uncovers the crucial property of sequential auctions that leads to inefficiency: the ability to respond to deviations. If all auctions happened simultaneously, then the behavior of the additive bidders that we described above could not possibly be an equilibrium: one additive bidder, knowing that his additive competitor bids on all the competitive items, would simply deviate to outbid him on the competitive items and get a huge utility. However, because the items are sold sequentially, this deviation cannot be undertaken without consequence: the moment one of the additive bidders deviates to bidding on the competitive items, in all subsequent auctions the competitor will respond by bidding on subsequent competitive items, leading to zero utility for the remainder of the auctions. Moreover, this response need not be punitive, but is rather the only rational response once the auction has left the equilibrium path (since the additive bidders know that there is no way to obtain positive utility in subsequent auctions). Thus, in a sequential auction, an additive player can only extract utility from at most one competitive item, which is not sufficient to counterbalance the resulting utility-loss due to the increased competition on the last non-competitive item.

The Lower Bound. We now proceed with a formal proof of Theorem 4.1. Consider an instance with additive players, unit-demand players and items. Denote with the two additive players and with the unit-demand players. Also denote the items with . The valuations of the additive players are represented by the following table of , where is an arbitrarily small constant:

In addition the unit-demand valuations for the players are given by the table of that follows (an empty entry corresponds to a valuation), though now a valuation of a player when getting a set is :

The constants are chosen to satisfy the following condition:

(1)

Note that, by taking to be arbitrarily small, we can take each to be arbitrarily small as well.

In the optimal allocation, player gets all the items and , player gets and player gets . The resulting social welfare is . We assume that the auctions take place in the order depicted in the valuation tables: . We will show that there is a subgame perfect equilibrium for this auction instance such that the unit-demand players win all the items . Specifically, player wins item , player wins , player wins , and player wins , resulting in a social welfare of . Taking sufficiently small, this welfare is at most . This will establish that the price of anarchy for this instance is at least , establishing Theorem 4.1. Furthermore, we will show that this subgame perfect equilibrium is natural, in the sense that it survives iterated deletion of weakly dominated strategies.

The intuition is the following: after the first auctions have been sold, player has to decide if he will target (and win) item , or if he will instead target items and/or . If he targets item , he competes with player and afterwards lets players and win items for free. This decision of player depends on whether player has won item , which in turn depends on the outcomes of the first auctions. In particular, player can win item only if player has won item . In turn, can win only if has won item and so on. Hence, it will turn out that in order for to want to target item , it must be that each item is sold to bidder . Thus, if either player or acquires any of the items , they will be guaranteed to obtain low utility on items and . This will lead them to bidding truthfully on all subsequent auctions, leading to a severe drop in utility gained from future auctions.

In the remainder of this section, we provide a more formal analysis of the equilibrium in this auction instance. We begin by examining what happens in the last three auctions of and , conditional on the outcomes of the first auctions. We first examine the outcome of auctions conditional on the outcome of auction :

  • Case 1: has won

    Player has marginal value of for item . Hence, he is willing to bid at most on item .

    Player knows that if he loses then in the subgame perfect equilibrium in that subgame he will bid on and and lose. Thus he expects no utility from the future if he loses . Thus he is willing to pay at most for item .

    Since by assumption (1) , player will win at a price of . Then players will win and for free. Thus the utilities in this case from this subgame are: , , , .

  • Case 2: has lost

    Player has marginal value of for item . Hence, he is willing to bid at most on item .

    Player performs the exact same thinking as in the previous case and thereby is willing to bid at most for item .

    Thus in this case will win item at a price of . Then, as predicted, will bid on and and lose. Thus the utilities of the players in this case are: , , , .

Now we focus on the auction of item . As was explained in Paes Leme et al. [19] this auction will be an auction with externalities where each player has a different utility for each different winner outcome. This utilities can be concisely expressed in a table of ’s where is the value of player when player wins. The only players that potentially have any incentive to bid on item are . The following table summarizes their values for each possible winner outcome of auction as was calculated in the previous case-analysis (we point that in the diagonal we also add the actual value that a player acquires from item to his future utility conditional on winning ) .

For example, player obtains utility if player wins item . We see from the table that, at this auction, everyone except achieves their maximum value when wins the auction. Player has value for winning the auction only if he hasn’t won . In addition, since , if hasn’t won then he can definitely outbid on and therefore has no chance of winning the auction of . As we now show, this implies that there is a unique equilibrium of the auction conditioning on whether or not has won :

  • Case 1: If has won then he has no value for . There exists an equilibrium in undominated strategies where and all players will bid , while bids . In fact this is in some sense the most natural equilibrium since it yields the highest utility for and . In this case the utility of the players from auctions and onward will be: , , , , .

  • Case 2: If has lost , then he has value of for . Hence, has no chance of winning item . Thus, the unique equilibrium that survives elimination of weakly dominated strategies in this case is for player to bid , for player to bid , for player to bid , for player to bid and for player to bid . In this case the utility of the players from auctions and on will be: , , , , .

Using similar reasoning we deduce that player can win only if has won . If at any point some does not win then players and know that from that point onward no can win auction , and therefore they will get only utility from . Thus there will be no reason for players and to allow unit-demand players to continue to win items, and thus the only equilibrium strategies from that point on will be for to bid on each of and to bid . This will lead to player to get utility from each auction for items , and player to get no utility from these auctions. Thus, at any point in the auction, it is an equilibrium for players and to allow the unit demand player to win auction conditional on the fact that they have allowed all previous unit-demand bidders to win. In particular, in the first auction, it is an equilibrium for players and to allow player to win. We conclude that the strategy profile we described is a subgame perfect equilibrium for this auction instance. This completes the proof of Theorem 4.1

Finally, as discussed throughout our analysis, the equilibrium described above survives IEWDS. The reason is that, for every item and bidder , the proposed equilibrium strategy for bidder does not require that he bid more than his value for item less his utility in the continuation game subject to not winning item . As discussed in Paes Leme et al. [19], this property guarantees that no player is playing a weakly dominated strategy.

5 Extensions of the Lower Bound

We now provide some reinterpretations and extensions of our lower bound from the previous section, to show that linear inefficiency can occur under several important classes of valuations.

Gross Substitutes.

Since the class of gross substitutes valuations includes all additive and unit-demand valuations, the example from the previous section immediately implies a linear price of anarchy for gross substitutes valuations.

Budget-Additive.

A valuation is budget additive if it can be written in the form . As it turns out, in the example in the previous section all valuations are budget additive. The additive players can be thought of as having infinite budget. Each of the unit-demand players for can be thought as budget-additive with a budget of and value for items and and for everything else. Player has budget of and additive value of for , for and for everything else. Player has budget and additive value of for each of and for everything else. Therefore the analysis in the previous section holds even for budget-additive valuations.

Additive valuations with budget constraints on payments.

We show that the same analysis can be applied to a setting in which each player has an additive valuation as well as a hard budget constraint on his payment. That is, his utility is quasi-linear as long as his payment is below , but becomes minus infinity if he pays more than . Formally, if a player receives a set and pays total price then his utility is if , or otherwise.

We will adapt the example from the previous section to the setting of budget constraints in a manner similar to the case of budget-additive valuations. Specifically, we set the budgets of the players as in the budget-additive case described above, but we treat them as payment budgets rather than a cap on valuations.

We need to be slightly careful in our analysis under this adaptation, since it doesn’t only matter whether a player won or lost an item, but also at which price. Specifically, the equilibrium will alter slightly. The additive bidders, apart from letting bidder win , will also have to make him pay enough so that he has no remaining budget with which to win the subsequent item .

For player , we know that his budget is indeed almost exhausted at auction whenever he wins, since player has a substantial value. Thus for auction no change in the equilibrium analysis takes place. However, when examining auction , if we consider the same equilibrium as in the previous section, then player pays nothing and thus still has all his budget to bid on and win it. It is in the interest of the additive bidders to ensure that not only wins, but also pays at least , so that he doesn’t have enough budget to win item .

Player knows that if he loses the auction for item then he can use his budget to get utility of from winning . If he wins for a price of then he gets no utility from the future and instead gets a utility of from winning . Assuming that , player is willing to pay more than to win auction . Thus, if we assume , the additive players can bid enough on item that player will win it at some price above , which will then result in winning and the additive bidders getting utility from and . A similar analysis holds for the auction of each item , for : the additive players need to make sure that each bidder wins , and also pays enough so that he doesn’t have enough budget to tilt player on getting his next item rather than . However, observe that if player loses auction , then subsequently the additive players will switch to winning all the remaining items, since there is no hope to make the unit-demand bidders win their items; so it is in the interest of each player to accept any price up to and therefore the additive players can completely exhaust his budget. With this change in the equilibrium strategies, our analysis in the previous section carries over, and we conclude that the price of anarchy in this instance is .

6 The Impact of Item Ordering

Our lower bound establishes that if items are sold sequentially, then arbitrarily inefficient outcomes can result at equilibrium even when all agents have gross substitutes valuations. The constructions depend on the items being sold in an arbitrary order. A natural question arises: does there always exist an order over the items such that the resulting outcome is efficient, or approximately efficient?

In this section we discuss this problem in the context of unit-demand bidders. Recall that, for unit-demand bidders, selling items in an arbitrary order always results in an outcome that achieves at least half of the optimal social welfare. Additionally, it is known by [19] that if any order is allowed then the unique subgame-perfect equilibrium that survives IEWDS can be inefficient, achieving only a -approximation. This lower bound of holds even for the special case of single-valued unit-demand bidders, where each player has a single value for getting one item from some interest set . We conjecture that, for the case of single-valued unit-demand bidders, if the auctioneer can choose the order in which the objects are sold, then it is possible to recover the optimal welfare at all natural equilibria. Indeed, we make a stronger conjecture: there exists an order in which the VCG outcome (allocation and payments) occurs at equilibrium.

Conjecture 1

For every instance of single-valued unit-demand bidders, there exists an order over the items such that the corresponding sequential auction admits a subgame perfect equilibrium that survives IEWDS and that replicates the VCG outcome.

Observe that such a result cannot hold for both additive and unit-demand bidders as is portrayed by our simple example in Section 3, where all items are identical and hence, under any ordering, the unique subgame-perfect equilibrium that survivies IEWDS is inefficient. Our conjecture also stems from the fact that for the case of single-valued unit-demand bidders the optimization problem is a matroid optimization problem. It is known by [19] that a form of sequential cut auction for matroids always leads to a VCG outcome. The difference is that sequential item-auctions do not correspond to auctions across cuts of the matroid. However, it is feasible that under some ordering the same behavior as in a sequential cut auction will be implemented.

As progress toward this conjecture, we will present a subset of item orderings, the augmenting path orderings, which we believe always contains an ordering that satisfies Conjecture 1. For instance, we show in Appendix B that the lower bound of [19] breaks if we only allow augmenting path orderings. We leave open the question of whether one of these orderings always yields a VCG outcome.

6.1 A Class of Orderings

Consider a profile of single-valued unit-demand valuations. Let denote the VCG allocation (i.e., is the item allocated to bidder ). We also write to denote the VCG allocation when bidder is excluded. For each , the allocations and define a directed bipartite graph between players and objects, where there is an edge between player and item if but , and there is an edge from item to player if but . It is known that, for each player , this graph is always a directed path from player to some other player ; this is the augmenting path for player and player is the price setter of player , i.e. the VCG price of player is . With no loss of generality we assume that every player has a price setter .

Given a welfare-optimal matching , that matches each player to an item , consider the following forest construction. Consider all price setters in decreasing value order. For each price setter , we will create a tree and add it to the forest, as follows. Consider all the items that are in the interest set of , , that are not yet in the forest. Add each such item to the tree as a child of player . Next, from each such item , consider its optimally matched player and add this player to the tree as a child of . For each player that was added, consider all items that are in the interest set of , , that are not yet in the forest, and add each of these items to the tree as a child of . We continue this process, which is essentially a breadth-first traversal of the set of items, until there is no new item to be added.

The above process creates a forest that contains a node for each item, for each player that is allocated an item in the optimal allocation, and for each price setter. Additionally, each player belongs to the tree rooted at his price setter and his unique path in the tree to the price setter is an augmenting path in the initial bipartite graph. The reasoning is as follows: each tree contains all possible alternating paths ending at the price-setter, except alternating paths that contain items and players who have been included in the tree of a price setter with larger value. Since a player’s price setter is the largest unallocated player with which he is connected, through an alternating path, the claim follows.

We will refer to the above forest as the augmenting path graph . Given an augmenting path graph , a post-order item traversal of is a depth-first, post-order traversal of the nodes of , restricted to the nodes corresponding to items and rooted at price setters. Note that this is an ordering over the items in the auction. We also assume that trees are traversed in decreasing order of price-setters. Also note that this order is not necessarily unique, as it does not specify the order in which the children of a given node should be traversed.

Definition 6.1

The set of augmenting path orderings of the items is the set of orderings corresponding to post-order item traversals of .

Our (refined) conjecture is that, for every instance of single-valued unit-demand bidders, there exists an augmenting path ordering such that the corresponding sequential auction admits a subgame perfect equilibrium that replicates the VCG outcome. As an example, we show in Appendix B that this conjecture holds for the lower bound example from [19]. We also show in Appendix C that it is not true that all augmenting path orderings lead to efficient outcomes at equilibrium: there are examples in which multiple augmenting path orderings exist, and some orderings lead to inefficient outcomes at equilibrium.

Appendix A Iterated Elimination of Weakly Dominated Strategies

When considering subgame perfect equilibria of sequential item auctions, we wish to restrict our attention to “natural” equilibria, that exclude (for example) dominated overbidding strategies. We therefore consider a natural and well-studied refinement of the set of subgame perfect equilibria: those that survive iterated elimination of weakly dominated strategies (IEWDS). A strategy is weakly dominated by a strategy if, for every profile of other players’ strategies , we have , and moreover there exists some such that . We can now define what it means for a strategy profile to survive iterated elimination of weakly dominated strategies.

Definition 1

Given an -player game defined by strategy sets and utilities we define a valid procedure for eliminating weakly dominated strategies as a sequence such that for each there is an such that for , , and for all there is some such that for all and the inequality is strict for at least one . We say that a strategy profile survives iterated elimination of weakly dominated strategies (IEWDS) if, for any valid procedure , .

Appendix B Augmenting Path Orderings: An Example

In [19], it was shown that there exist single-valued unit-demand auctions in which inefficient outcomes can occur when items are sold sequentially in an arbitrary order. In this section we motivate that augmenting path ordering by showing that, for this example, the efficient outcome occurs when the items are sold according to their augmenting path order.

We begin by recalling the example. There are three items, , and 4 players . We fix an arbitrarily small constant . Recall that the valuation of each player is specified by a real value and a set of items of interest; the player then has value for any item in and value for any other item. The valuations in our example are given by:

  • and ,

  • and ,

  • and , and

  • and .

The welfare-optimal allocation is , for a social welfare of . The VCG prices are . Note that, in the terminology of Section 6, player is the price-setter for each of the other players. In [19] it is shown that if the items are auctioned in the order , then the unique subgame perfect equilibrium that survives IEWDS leads to an inefficient outcome.

What are the augmenting path orderings in this example? In this example, the augmenting path graph is a line, given by nodes in that sequence. There is therefore a unique augmenting path ordering over the items: the order .

We can now solve for the subgame perfect equilibrium of the auction when items are sold in this order. We do so by analyzing the item auctions in reverse order. When item is sold, the outcome depends on whether or not player won item : if so, player will win item for a price of , yielding ; if not, then player will win item for a price of , yielding and . This allows us to determine the outcome of the auction for item : because player knows that she can win item for a price of , she is willing to bid no more than on item . Thus, if player did not previously win item , then player can win item with a bid of , yielding . This ultimately allows us to determine the outcome of the first auction, the auction for item . Because player knows that he can win item for a price of , she is willing to bid no more than on item . Since player obtains positive utility only if she wins item , she is willing to bid as much as on item . We therefore have that player will choose to win item with a bid of , obtaining utility . Applying our analysis of the subsequent auctions, we conclude that bidder will win item for a price of , and then bidder will win for a price of .

Note that this subgame perfect equilibrium, which is the unique equilibrium in undominated strategies, precisely implements the VCG outcome. Moreover, our analysis extends easily to other values of , and , as long as they are all greater than .

Appendix C Not all Augmenting Path Orderings lead to Efficiency

We now show that if the augmenting path graph is not a line, then some augmenting path orderings do not result in an efficient outcome, even if valuations are unit demand single-valued.

The example is as follows. There are items, say . There are players. Player wants all items and has value . Player wants only item and has value . Player wants item or and has value . Player want item or and has value .

In this example, the VCG outcome is , and the VCG prices are . The augmenting path graph is a tree with player at the root, each item a child of player , and each remaining player being the child of item . For this graph, every order over the items is an augmenting path order.

Suppose the items are sold in the order . In the VCG outcome, player obtains utility . In the sequential play corresponding to the VCG outcome, players and both bid their values on item . Consider the following deviation by player . When item is sold, he bids , causing player to win item . Item will sell next; players and will bid on it. Consider what would happen if player wins item : in this case, players and both bid their values on item , and hence player wins and player ends with utility . We conclude that player prefers to win item at any price less than , and hence will bid on item , winning it. Thus, when item is sold, only player places a non-zero bid, winning the item at price . We conclude that, after this deviation, player obtains utility , and therefore the VCG outcome is not a SPE.

Footnotes

  1. The equilibrium that we describe is, in some sense, the unique natural equilibrium: if we were to ask players to submit bids sequentially within each auction, rather than simultaneously, then there would be a unique equilibrium (solvable by backward induction), which is the equilibrium that we describe.
  2. In fact, optimality is always achieved when all bidders are additive, in general.

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