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