Arbitrage in Combinatorial Exchanges

Andrew Gilpin and Tuomas SandholmCarnegie Mellon University

Computer Science Department

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Combinatorial exchanges

• Trading mechanism for bundles of items

• Expressive preferences– Complementarity, substitutability

• More efficiency compared to traditional exchanges

• Examples: FCC, BondConnect

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Other combinatorial exchange work

• Clearing problem is NP-complete– Much harder than combinatorial auctions in practice

– Reasonable problem sizes solved with MIP and special-purpose algorithms [Sandholm et al]

– Still active research area

• Mechanism design [Parkes, Kalagnanam, Eso]– Designing rules so that exchange achieves various

economic and strategic goals

• Preference elicitation [Smith, Sandholm, Simmons]

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Uncovered additional problem: Arbitrage

• Arbitrage is a risk-free profit opportunity• Agents have endowment of money and

items, and wish to increase their utility by trading

• How well can an agent without any endowment do?– Where are the free lunches in combinatorial

exchanges?

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Related research: Arbitrage in frictional markets

• Frictional markets [Deng et al]– Assets traded in integer quantities– Max limit on assets traded at a fixed price

• Many theories of finance assume no arbitrage opportunity

• But, computing arbitrage opportunities in frictional markets is NP-complete

• What about combinatorial markets?

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Outline• Model• Existence

– Possibility– Impossibility

• Curtailing arbitrage• Detecting arbitraging bids• Generating arbitraging bids• Side constraints• Conclusions

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Model

• M = {1,…,m} items for sale

• Combinatorial bid is tuple: = demand of item i (negative means supply) = price for bid j (negative means ask)

• We assume OR bidding language– As we will see later, this is WLOG

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Clearing problem

• Maximize objective f(x)– Surplus, unit volume, trade volume

• Such that supply meets demand– With no free disposal, supply = demand

• All 3 x 2 = 6 problems are NP-complete

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Arbitraging bids in a combinatorial exchange

• Arbitrage is a risk-free profit opportunity– So price on bid is negative

• Agent has no endowment– Bid only demands, no supply

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Impossibility of arbitrage

• Theorem. No arbitrage opportunity in surplus-maximizing combinatorial exchange with free disposal

• Proof. Suppose there is. Consider allocation without arbitraging bid– Supply still meets demand (arbitraging bid does

not supply anything)– Surplus is greater (arbitraging bid has negative

price). Contradiction

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Possibility of arbitrage in all 5 other settings

• M = {1, 2}• B1 = {(-1,0), -8} (“sell 1, ask $8”)• B2 = {(1,-1), 10} (“buy 1, sell 2, pay $10”)

– With no free disposal, this does not clear

• B3 = {(0,1), -1} (“buy 2, ask $1”)– Now the exchange clears

• Same example works for unit/trade volume maximizing exchanges with & without free disposal

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Even in settings where arbitrage is possible, it is not possible in every instance

• Consider surplus-maximization, no free disposal

• B1 = {(-1,0),-8} (“sell 1, ask $8”)

• B2 = {(1,-1),10} (“buy 1, sell 2, pay $10”)

• B3 = {(0,1), 2} (“buy 2, pay $2”)

• No arbitrage opportunity exists

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Possibility of arbitrage: Summary

Objective Free Disposal No Free Disposal

Surplus Impossible Sometimes possible

Unit volume Sometimes possible Sometimes possible

Trade volume Sometimes possible Sometimes possible

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Curtailing arbitrage opportunities

• Unit/trade volume-maximizing exchanges ignore prices

• Consider two bids:– B1 = {(1,0), 5} (“buy 1, pay $5”)– B2 = {(1,0), -5} (“buy 1, ask $5”)

• In a unit/trade volume-maximizing exchange, these bids are equivalent

• Can we do something better?

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Curtailing arbitrage opportunities…

• Run original clearing problem first

• Then, run surplus-maximizing clearing with unit/trade volume constrained to maximum

• This prevents situation from previous slide from occurring

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Detecting arbitraging bids

• Arbitraging bid can be detected trivially– Simply check for arbitrage conditions

• Theorem. Determining whether a new arbitrage-attempting bid is in an optimal allocation is NP-complete– even if given the optimal allocation before that bid was

submitted– Proof. Via reduction from SUBSET SUM– Good news: Hard for arbitrager to generate-and-test

arbitrage-attempting bids

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Relationship between feedback to bidders and arbitrage

• Feedback– NONE– OWN-WINNING-BIDS– ALL-WINNING-BIDS– ALL-BIDS

• Feedback ALL-BIDS provides enough information to bidders for them to arbitrage

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Generating arbitraging bids (for any setting except surplus-maximization with free disposal)

• If all bids are for integer quantities, arbitrager can simply submit 1-unit 1-item demand bids (of price )

• Otherwise, arbitraging bids can be computed using an optimization (related to clearing problem)– Item quantities are variables

– Problem is to find a bid price and demand bundle such that the bid is arbitraging:

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Side constraints

• Recall: Arbitrage impossible in surplus-maximization with free disposal

• Exchange administrator may place side constraints on the allocation, e.g.:– volume/capacity constraints– min/max winner constraints

• With certain side constraints, arbitrage becomes possible …

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Side constraints: Example

• Side constraint: Minimum of 3 winners• Suppose:

– Only two bidders have submitted bids

– Without side constraint, exchange clears with surplus S

• Third bidder could place arbitraging bid with price at least –S

• Thus, arbitrage possible in a surplus-maximizing CE with free disposal and side constraints

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Bidding languages

• So far we have assumed OR bidding language

• All results hold for XOR, OR-of-XORs, XOR-of-ORs, OR*– Does not hurt since OR is special case– Does not help since arbitraging bids do not

need to express substitutability

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Conclusions• Studied arbitrage in combinatorial exchanges

– Surplus-maximizing, free disposal: Arbitrage impossible– All 5 other settings: Arbitrage sometimes possible

• Introduced combinatorial exchange mechanism that eliminates particularly undesirable form of arbitrage

• Arbitraging bids can be detected trivially• Determining whether a given arbitrage-attempting bid

arbitrages is NP-complete (makes generate-and-test hard)• Giving all bids as feedback to bidders supports arbitrage• If demand quantities are integers, easy to generate a herd of

bids that yields arbitrage– If not, arbitrage is an integer program

• Side constraints can give rise to arbitrage opportunities even in surplus-maximization with free disposal

• The usual logical bidding languages do not affect arbitrage possibilities