Arbitrage in Combinatorial Exchanges

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

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

3. 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. 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. 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. Outline • Model • Existence • Possibility • Impossibility • Curtailing arbitrage • Detecting arbitraging bids • Generating arbitraging bids • Side constraints • Conclusions

7. 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. 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. 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. 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. 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. 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. Possibility of arbitrage: Summary

14. 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. 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. 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. 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. 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. 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. 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. 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. 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