Arbitrage in
Combinatorial Exchanges
Andrew Gilpin and Tuomas Sandholm
Carnegie Mellon University
Computer Science Department
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 specialpurpose 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
6 / 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
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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 arbitrageattempting bid is in an optimal allocation is NPcomplete
– 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 surplusmaximization 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
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