1. Arbitrage in Combinatorial Exchanges Andrew Gilpin and Tuomas Sandholm Carnegie Mellon University Computer Science Department

2. 2 / 22 Combinatorial exchanges Trading mechanism for bundles of items Expressive preferences –Complementarity, substitutability More efficiency compared to traditional exchanges Examples: FCC, BondConnect

3. 3 / 22 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]

4. 4 / 22 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?

5. 5 / 22 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?

6. 6 / 22 Outline Model Existence –Possibility –Impossibility Curtailing arbitrage Detecting arbitraging bids Generating arbitraging bids Side constraints Conclusions

7. 7 / 22 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

8. 8 / 22 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

9. 9 / 22 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

10. 10 / 22 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

11. 11 / 22 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

12. 12 / 22 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

13. 13 / 22 Possibility of arbitrage: Summary ObjectiveFree DisposalNo Free Disposal SurplusImpossibleSometimes possible Unit volumeSometimes possible Trade volumeSometimes possible

14. 14 / 22 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?

15. 15 / 22 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

16. 16 / 22 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

17. 17 / 22 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

18. 18 / 22 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:

19. 19 / 22 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 …

20. 20 / 22 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

21. 21 / 22 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

22. 22 / 22 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