An Options-Based Method to Solve the Composability Problem in Sequential Auctions

. Current auctions often expose bidding agents to two dif(cid:2)cult, yet common, problems. First, bidding agents often have the opportunity to participate in successive auctions selling the same good, with no dominant bidding strategy in any single auction. Second, bidding agents often need to acquire a bundle of goods by bidding in multiple auctions, again with no dominant bidding strategy in any single auction. This paper introduces an options-based infrastructure that respects the autonomy of individual sellers but still enables bidders to utilize a dominant, truthful strategy across multiple auctions.


Introduction
Many authors (e.g., [8]) have written about a future in which commerce is mediated by automated trading agents. Yet, we believe that one leading place of resistance is in the lack of optimal bidding strategies for any but the simplest of market designs. Although it is popular to appeal to computational mechanism design [5], and try to design truthful auctions to address this problem, it is nevertheless clear that a single truthful mechanism cannot exist for all transactions in which an agent has an interest. Somewhere, at some point, there must be boundaries between mechanisms [20].
This work proposes a new options-based market infrastructure, that can enable simple yet optimal bidding strategies, while retaining the seller autonomy that is the defining feature of the most successful of today's electronic markets. Although eBay acts as an intermediary of sorts, eBay does not gather up the goods for sale by multiple sellers and run them within a single coordinated event. Rather, eBay is an open environment in which each seller chooses: a) when to bring a good to market; and b) the kind of auction to use. Buyers then pick-and-choose across auctions, before submitting bids and making purchases.
One problem encountered in environments like eBay is the exposure problem. People would like to acquire multiple items, but may end up only holding a subset of those items at the end. For example, imagine Alice would like to buy both Peanut Butter (¢ ¤ £ ) and Jelly (¥ ), but has to participate in two different auctions in order to acquire both items. Alice may bid enough to win the ). In an eBay environment, the presence of millions of different goods (with an exponentially larger number of possible bundles of goods) makes the design of a single mechanism impractical. Second, CAs do not resolve temporal issues, assuming instead that all agents are present at the same time. Third, CAs assume an unrealistic market scope, with one market-maker able to control and bring together all participants into a single market.
Retail stores have customers that face similar strategic problems as the composability problem, and they have devised different policies to alleviate the problems that their customers face. Return policies alleviate the exposure problem by allowing customers to return goods at the purchase price. Price matching alleviates the multiple copies problem by allowing agents to receive from sellers after product purchase the difference between the price paid for a good and a lower price found elsewhere else for the same good. These two retail policies provide the basis for the scheme proposed in this paper. In particular, we propose an options-based infrastructure to address the composability problem.
To participate in the options scheme, a seller must agree to sell an option for her good, which will ultimately lead to either a sale of the good, or (if the option is not exercised) going back to the market and offering another option on the same good. The process by which a seller sells options and leaves/returns to market can be made easier via a proxy, whereby a seller may tell a proxy her patience and good for sale, after which the proxy could then sell an option for that good at auction, observe the status of the sold option, and resell another option if the initial option is not exercised and the seller's patience has yet to expire.
Buyers can collect portfolios of options before deciding which to exercise. We provide buyers with mandatory proxy agents, that carefully control the number of outstanding options that each buyer can hold, and yet still follow one of the dominant bidding strategies that an agent could follow if there was no proxy in the system. A buyer reports her value and patience to a proxy agent upon arrival, and then agrees to let her proxy: a) bid in auctions to acquire options; b) exercise options to maximize reported utility once the buyer's patience is expired. The proxy agents are essential to prevent a buyer from obtaining options on which they have no intention of exercising. The options-based protocol makes truthful and immediate revelation to a proxy a dominant strategy for buyers, whatever the future auction dynamics.
A benefit for sellers as demonstrated through simulation is that the options-based protocol maintains a market even with buyers with complementary values for goods. In comparison, we show that a traditional market can fail, because it quickly becomes difficult for buyers to participate in the market without becoming exposed to partial bundles and losing money. Thus, the options-based scheme has appeal to both buyers and sellers.

Related Work
The composability problem was previously observed in Wellman & Wurman [20], in the context of a discussion about the boundaries that must inevitably exist between mechanisms. This theme was continued by Parkes [14] and Ng et al. [13] in the context of strategyproof computing, in which the goal is to promote the deployment of strategyproof mechanisms within an open and shared infrastructure. The problem has often been identified in the context of simultaneous ascending price auctions, where it is termed the exposure problem [4].
Previous work to address the problem has considered two different directions. First, one can change the mechanism to define an expressive bidding language and incentivecompatibility. This is the approach taken in the work on combinatorial auctions (see Rothkopf et al. [18]). Second, one can attempt to provide agents with smarter bidding strategies. This is the approach taken in the work of Boutilier et al. [2], Byde et al. [3], Anthony & Jennings [1], and Reeves et al. [7]. Unfortunately, it seems hard to design artificial agents with equilibrium bidding strategies, even for a simple simultaneous ascending price auction.
Iwasaki et al. [10] have considered options in the context of a single, monolithic, auction design to help bidding agents with marginal-increasing values avoid exposure in a multi-unit homogeneous item auction problem. Sandholm & Lesser [19] have considered options in the form of leveled commitment contracts for facilitating multi-way recontracting in a completely decentralized market place. Rothkopf & Engelbrecht-Wiggans [17] discuss the advantages associated with the use of options.
Recent work of Porter et al. [16] has considered auctions with uncertainty in an agent's ability to successfully complete a task. As in our work, there is uncertainty for a seller, in their setting due to whether or not a task will be performed. The chief difference is that bidders in their model know their fault probabilities, and the authors can design a mechanism around the revelation of this information.
Finally, a recent direction taken in computational mechanism design is that of online mechanisms [15] and online auctions [11,9], in which agents can dynamically arrive and depart across time. We leverage a price-based characterization in Hajiaghayi et al. [9] to provide a dominant-strategy equilibrium for buyers within our options-based protocol, creating a decentralized, truthful, option-based implementation of an online combinatorial auction.

The Composability Problem
To illustrate the composability problem, consider the following simple example in which a bidder does not have a dominant strategy equilibrium even though the individual auctions in the world are strategyproof.

The Model
Consider a world with goods, These are the goods that, when taken with some combination of other goods, have non-zero value to the agent. Buyers are indifferent between receiving a bundle of goods at any time before and up to her departure time. Each seller, 4 5 2 , sells a single item f g 4 x 0 . 1 All individual auctions in our model will therefore be for a single item. Seller has an arrival time,

Y 4 Ì
, and a departure time , when they will leave the system if no agent has yet obtained the right to purchase their good.
Agents arrive or leave at the end of each period, but multiple auctions can be sequenced within a period. Let ! denote this sequence of auctions in period , each one associated with a single seller. From buyer 3 ' s perspective, let 8 denote the sequence of auctions that occur during the time in which buyer arrives in period 1 and departs in period 3, then We require that each individual auction is strategyproof (SP). Following Parkes [14] we term this local strategyproofness, to emphasize that it does not imply that a buyer has a dominant strategy when bidding across multiple such auctions. The utility to a buyer, given goods @ and price , is defined as An individual auction can be defined in terms of an allocation rule, " A d e ( and a payment rule f " g d e ( , given bids , is a dominant bidding strategy when . For instance, a single-item Vickrey auction is locally-SP for all agents that will bid only in that auction.
When facing a sequence of auctions, a bidding strategy, d 8 for buyer 3 defines the bid that the agent will make in each auction, and can be contingent on: i) her own value; ii) her beliefs about other agents; iii) the outcomes and feedback from earlier auctions.

Definition 2 (The composability problem). The composability problem exists for an agent facing a sequence of auctions 8 , when each auction in 8 is locally-SP, but the agent does not have a dominant bidding strategy across the sequence of auctions.
In fact, the composability problem exists more often than not! In what follows, we assume that all goods in an agent's valuation function are available in 8 , and that each auction is locally-SP. All proofs are omitted in the interest of space. First, consider a single-minded buyer, and let Proof omitted for space. The effect of multiple auctions is that the agent must anticipate the level of competition, and prices, in future auctions when deciding how to bid. For instance, an agent that wants a single item and faces a sequence of Vickrey auctions does not have a dominant bidding strategy, but would prefer to bid in the auction with the lowest second price.
Next, we can consider an agent that demands multiple disjoint bundles Proof omitted for space. We can also consider an agent with a general valuation that cannot be expressed as an additive value across disjoint bundles, which precludes a single-minded agent.

Proposition 3. An agent with a general valuation faces the composability problem whenever it faces two or more interesting auctions.
Proof omitted for space. Here, there must exist bundles that are either substitutes or complements. If the bundles are substitutes, an agent faces the problem of determining which bundle to pursue (analogous to the problem an agent faces when the same single item is sold at multiple auctions). If the bundles are complements, an agent can face the exposure problem (analogous to when a single bundle contains multiple items).

The Opportunity to Use Options
An option is a right to acquire a good at a certain price, called the exercise price. For instance, Alice may obtain from Bob the right to buy ¢ ¤ £ from him at an exercise price of . What makes options unique is that the right to purchase a good at an exercise price does not imply the obligation to purchase a good at an exercise price. Therefore, when Alice obtains an option from Bob, Bob is not guaranteed that Alice will actually exercise the option at the exercise price and obtain the good. This flexibility makes options useful in addressing the composability problem. Buyers can put together a collection of options on goods, and then decide whether to exercise each option.
Options are typically sold, obtained at a price called the option price. However, options obtained at a non-zero option price can not generally support a simple dominant bidding strategy, as an agent must compute the expected value of an option [6] to justify the cost. This computation requires a model of the future, which in our setting requires a model of the bidding strategies and the values of other bidders. This is the very reasoning that we are trying to avoid by introducing options! Instead, we consider costless options, where the option price is zero. This will require some care.
The basic problem arises because agents are always (weakly) better off with an option than without an option, whatever its exercise price, because an agent can always choose for free not to exercise an option won. Therefore, an agent would be interested in obtaining a costless option at any exercise price (including infinity), subsequently choosing to exercise the option only if doing so would result in a gain of surplus. However, multiple bidders pursuing options with no intention of exercising them could cause market efficiency to unravel. We address this issue through mandatory proxy agents, which intermediate between buyers and the market.

Auctions for Options
In our scheme, sellers run an auction for costless options on goods, and buyers bid through mandatory proxy agents. These proxy agents are critical to addressing the potential for an inefficient allocation of options through hoarding. Proxy agents, coupled with auctions for options, make it a buyer's dominant strategy to truthfully reveal her valuation and patience. Proxies follow a dominant bidding strategy for a buyer (by bidding at a value high enough that no higher bid could make the agent strictly better off), but restrict a buyer from pursuing options on which it is indifferent, such as a second option for a good when only one instance of the good is desired, or an option with an exercise price that could never be exercised for positive surplus. We now define the two main elements of our market: Seller Auctions. Each seller sells a costless option in a Vickrey auction. The option is issued to the highest bidder, with an exercise price equal to the second-highest bid, and is set to expire at the end of the buyer's patience. Sellers also agree, by joining the market, to allow the proxy representing a winning buyer to adjust downwards the exercise price if the proxy discovers that it could have achieved a better price by waiting to bid in a later auction for the same good (i.e., sellers agree to price match their competitors). Sellers can run additional auctions if the options are returned.

Proxy Agents. Each buyer
3 must submit to a proxy an expressive language bid, u 7 8 , reporting values for desired bundles. Each buyer also must submit a departure period, u a 8 , to the proxy. The proxy computes the maximal value for each item f desired by The proxy bids this price in any auction for item f , until it holds an option on this item. At that point, the proxy tracks future auctions on that item, determines what the world would look like if it had delayed its entry into the market until that later auction, and will reduce the exercise price of the option it has if it discovers that it could have secured a lower price by waiting to bid in that later auction. 3 The proxy determines this information by asking each future auction to report the identities (can be pseudonymous) of the winner and second-highest winner, together with their bid prices. The identities are necessary because they are used by the agent when creating its view of the world had it decided to delay its entry. 4 Finally, at the end of the buyer's reported patience u a 8 , the proxy exercises options to maximize reported value used to exercise options. Without this, agents could indifferently acquire options with exercise prices too high to ever be exercised. The proxy 2 While sellers of the same item type may not have different reserve prices for their goods (due to potential conflicts in being able to price match), sellers may agree (or be required) to have a universal reserve price for each item type, p } . In such a scenario, bidding agents can incorporate this information into their bids for multi-item bundles because it provides a tighter lower-bound on the price; specifically, However, the proxy does not at any point acquire a second option for the good. Rather, it retains the single option it has been holding, but reduces its exercise price to the later price. 4 In particular, the proxy maintains a candidate agent, , for each item on which it holds an option. Agent is the agent still present in the market that is currently not allocated an option for , but would have been by now had the proxy delayed its entry. There may be no such candidate agent if the displaced winner leaves the market without winning in a subsequent auction, at which point the state of the market looking forward is unaffected by wins, the exercise price for the option held by º is adjusted to the minimal of its current value and the second-highest price in this new auction, and the second-highest bidder in this new auction becoming the new candidate; else, the proxy's price is adjusted to the minimal of its current price and the highest bid price in this new auction. agent also ensures that no buyer can hold more than one option on each good, and can hold options on no goods outside its demand set. Without this, agents could indifferently obtain options that they have no intention of exercising. These two properties help to provide a well-functioning market. and Bob's proxy bids g » , resulting in Alice winning an option for the red hat with an exercise price of # » . On day two, a blue hat auction is held where Alice's proxy bids and Charlie's proxy bids 9 , resulting in Charlie winning an option for the blue hat with an exercise price of . At the end of day two, Alice's proxy exercises her red hat option and Charlie's proxy exercises his blue hat option.

Truthful Bidding to Proxy Agent
What remains to be shown is that it is a dominant strategy for the buyer to truthfully reveal her value and arrival and departure time to her proxy agent. The proof builds on the price-based characterization of time-strategyproof auctions in Hajiaghayi et al. [9].

Lemma 1. An online unit-demand auction is time and value strategyproof when:
it constructs a price function for the agent over time that is independent of the type reported by the agent, it allocates the good to the agent at the minimal price during period in the market, in a period no earlier than this minimal price, and only when the agent's value is higher than this minimal price.

Theorem 1. It is a dominant strategy for a buyer to truthfully reveal her valuation function and patience to her proxy agent in the options-based market.
Proof. (sketch) The options scheme constructs an agent-independent price schedule, to , whenever this price is less than the maximal value the agent could have for the item (given possible future prices). Overstating the value on a bundle can lead to the proxy holding an option on some item, f , at some price greater than it would ever want to pay. Understating the value can lead to the proxy missing a useful option on some item, f , and will not reduce the price otherwise. Strategies that misstate arrival and departure are not useful because reporting a later arrival or earlier departure can forfeit opportunities, while reporting a later departure, u a 8 y À a 8 , is not useful because the agent will not receive its goods until after a 8 . Thus, there is no useful manipulation of the final options on individual items, and finally the proxy makes a purchasing decision by looking at prices on options and exercising those that maximize reported utility.

Experiments
Up to this point we have focused solely on buyers. However, sellers can also benefit because the options-based scheme fixes the market failure that exists when buyers have complex values but face a sequence of auctions. The experimental results in this section demonstrate that there are many scenarios in which the average buyer surplus in a market without options is negative. In such a world, buyers would not enter the market to begin with (such a decision is not individually-rational) and there would be no market formation.
We simulate a simple market to better understand the economic effect of the options scheme, for both buyers and sellers. We construct values for buyers according to a quadratic method [12], which is parameterized with The value on the bundle is defined as . Each seller sells a single item, chosen uniformly at random from the set of all goods. We choose to model an identical reserve price for all goods and for all sellers, which is set to for each good f unless stated otherwise. 5 Buyer patience is set to 50, while seller patience is set to 100. We vary the buyer entry-rate and seller entry-rate to model different levels of supply and demand. We compare the options-based market with a market in which there is a sequence of Vickrey auctions for traditional goods. To model sell-side auctions in each round we choose to run Ñ auctions in each period . Upon arrival, sellers wait in a queue for their auction to be scheduled, on a first-come first-served basis. The rate, Ñ , is adjusted to keep the wait time, defined as the time that a seller needs to wait to have her auction scheduled, below 5 periods. 6 In the options world, a seller returns to the end of the queue if there are no bidders in her auction, or if the winner returns her option. In the non-options world, a seller only returns to the queue if she fails to sell her item.
Finally, we need a model of buyer strategies. In the options world, we assume that each buyer reports her true value to her proxy immediately upon arrival. In the nonoptions world, we need to adopt a bidding strategy for buyers, to provide a meaningful comparison with the performance of the options world. Ideally we would adopt an equilibrium bidding strategy, but this analysis is not available for such a complex game. Instead, we adopt a "sunk-aware" bidding strategy, following the ideas in Reeves et al. [7]. At any point in time an agent has purchased goods @ , for price ) Ò " f (  . Note that this can go negative, when a buyer pays more than her value for a bundle. We normalize in this way to remove dependencies on absolute values of goods in our empiric analysis. In the non-options world, when a buyer can fail to put together a complete bundle, we substitute v Ô 9 å , where å 6 § Â y 9 7 8 " A @ B ( , i.e. the value of the most-preferred bundle. Seller surplus is measured as the ratio of total revenue generated by all sellers divided by the total value of goods allocated to buyers. Losers are the percentage of buyers that leave the market with negative surplus. In some markets without options buyers can lose money, even when bidding conservatively. In this case, we also calculate the adjustedseller surplus, by factoring out any revenue that sellers were achieving from buyers that were losing money (which we would not expect in a sustainable equilibrium). . We also plot the percentage of buyers that are losers. Subplots (a)-(c) hold the seller entry rate fixed, whereas the seller entry rate in (d) is scaled with buyer entry rate at a 2:1 ratio. Figure 1 illustrates buyer and seller surplus against an increasing buyer entry-rate, with each subplot dedicated to a different structure " A Á Â Q $ Ã ³ ( for buyer valuations. We consider values of in subplots (a), (b) and (c) respectively, with the seller entry-rate set to 3, 6 and 12 in each scenario (this increases supply in line with increased buyer demand as the number of items Á demanded in each bundle increases). Figure 1 (a) demonstrates that the options and non-options world produce very similar results when the population has the most simple of valuations, with seller surplus increasing and buyer surplus decreasing as demand increases. Similar results were experimentally confirmed for all In Figure 1 (b) and (c) the non-options world "breaks" when demand gets too high because the composability problem becomes more challenging. The average buyer innot only would their income be lost to sellers, but also the prices paid by those people who remained would be lower due to the decreased competition. curs negative surplus and one would reasonably expect that buyers would not enter this market in the first place. On the other hand, buyer surplus in the options world remains positive, indicating that buyers would continue to enter the market. This suggests the existence of scenarios in which introducing options can create new markets. Further evidence for market breakdown in the non-options world can be found by considering the Losers rates. Figure 1 (c) shows that the Losers rates is near 0.5 when there is high demand, indicating that nearly half of the agents who are entering the market are losing surplus upon exit.
In Figure 1 (d), we consider , and scale the seller entry-rate continuously as we scale the buyer entry-rate, keeping the seller entry at twice the buyer entry. Whereas scaling the world degrades the buyer surplus in the non-options world to the point of being negative, seller surplus in the options world is steady and accompanied by positive buyer surplus. Figure 2 (a) fixes

Á § S
and increases the number of items in each bundle. We scale the buyer entry-rate from 16 to 8 to 5 to 4, while the seller entry-rate is fixed at 16, to keep supply and demand in the same proportion. The exposure problem is present in this scenario, and indeed we see a significant drop in buyer surplus and a rise in the percentage of buyers with negative surplus in the non-options world. Figure  2 (b) illustrates the effect of changing the reserve price on all items. Buyer entry-rate is 3, seller entry-rate is 9, and buyers demand 3 bundles, each with 3 items. A higher reserve price in the non-options world drives buyer surplus negative, and results in market failure. On the other hand, sellers in the options world are able to raise reserve prices to increase their surplus, and buyers can still manage to obtain positive surplus from the market even when reserve prices are set very high. Noteworthy, although the seller surplus is increasing in Figure 1, one should appreciate that this is a relative metric. In fact, we observe that the total seller surplus tends to decrease in the options world as buyer entry-rate grows (even though the normalized surplus remains quite flat). Less buyers complete their desired bundles and less buyers eventually exercise their options. Of course, we believe this is preferable to the complete market failure in the non-options world. However, while markets continue to form with options, trades are more infrequent as buy-side competition increases-which suggests that it would be interesting to explore additional methods, such as a prequalification stage, the use of stronger reserve prices, or the "throttling" of buyer entry-rate and pooling into separate markets. We reserve these topics for future study.

Conclusion
We introduced an options-based auction protocol to address the composability problem that exists when buyers with complex values must bid in sequences of simple auctions. Our approach combines costless options with proxy agents, which acquire, maintain, and exercise options on the agent's behalf and best interest. Simple trading agents have dominant bidding strategies in our options-based market, even though the markets remain fundamentally disintermediated. We believe that options-based markets may provide an interesting new class of market designs for eBay-like electronic markets.
Future work should aim to better situate this work within the context of the theory of strategyproof online auctions. Future work may also address and resolve the strategic problems facing sellers in this work. While it is not a dominant strategy for sellers to try and keep prices on their goods artificially high (as doing so may prevent options from being exercised if the prices maintained are at a prohibitively high level), it is true that straightforward truthful behavior may not always be in the best interest of sellers in the current model. Furthermore, an investigation of the role of false-name behavior [21] should also yield interesting results. While buyers do not want to engage in false-name behavior as multiple buyers, and sellers do not wish to engage in false-name behavior as multiple sellers, we believe there are manipulations for buyers pretending to be sellers and for sellers pretending to be buyers.
Additionally, future work on the empirical aspects of this project should aim to utilize better benchmarks when analyzing the model, including the use of real data. In particular, there are three areas where real data could be particularly helpful to this model. First, we believe there is ample opportunity for further exploration as to modeling the arrival of sellers and timing of auctions in this setting, perhaps using data from eBay as a foundation. Second, the high rates of buyers that are losing surplus in our simulation of buyers in the non-options model when demand is high is cause to believe that agents may follow a different bidding strategy than the one assumed here. Real world data can be of great assistance in helping to empirically determine what those strategies might be. Third, real world data can also help in developing accurate valuation models for some set of niche goods in an existing market.