Research Labs A Dynamic Pari-Mutuel Market for Hedging, Wagering, and Information Aggregation David M. Pennock Mike Dooley Previous Version Appears in EC’ 04, New York
Research Labs What is a pari-mutuel market? A B • E. g. horse racetrack style wagering • Two outcomes: A B • Wagers:
Research Labs What is a pari-mutuel market? A B • E. g. horse racetrack style wagering A • Two outcomes: B • Wagers:
Research Labs What is a pari-mutuel market? A B • E. g. horse racetrack style wagering A • Two outcomes: B • Wagers:
Research Labs What is a pari-mutuel market? A B • E. g. horse racetrack style wagering A • Two outcomes: B • 2 equivalent $ on B 8 1+ =$3 ways to consider $ on A 4 payment rule • refund + share of B • share of total $ 12 = = $3 $ on A 4
Research Labs What is a pari-mutuel market? • Before outcome is revealed, “odds” are reported, or the amount you would win per dollar if the betting ended now • Horse A: $1. 2 for $1; Horse B: $25 for $1; … etc. • Strong incentive to wait • • payoff determined by final odds; every $ is same Should wait for best info on outcome, odds No continuous information aggregation No notion of “buy low, sell high” ; no cash-out
Research Labs Dynamic pari-mutuel market Basic idea 1 1 1
Research Labs Dynamic pari-mutuel market Basic idea 1 1 0. 2 0. 4 1. 3 1. 6 0. 9 2 5 2. 3
Research Labs Dynamic pari-mutuel market Setup & Notation A 1 A 2. . . Ak • • • Outcomes: Prices (per share): Payoffs (per share): Money wagered on i: # shares purchased of i: Ai pi Pi Mi Si • Total money: T = j Mj • Share-weighted total: • “Other” money: • “Share-weighted other” $: W T-i W-i = j S j Mj = T - Mi = W - S i Mi
Research Labs What is a share? • Two alternatives; Share of • losing money only Winners get: original money refunded + equal share of losers’ money • all money Winners get equal share of all money • For standard PM, they’re equivalent • For DPM, they’re not
Research Labs How are prices set? • A price function pi(n) gives the instantaneous price of an infinitesimal additional share beyond the nth • Cost of buying n shares: 0 pi(n) dn • Different reasonable assumptions lead to different price functions n
Research Labs “All money” case • Payoffs: Pi=T/Si • Trader’s exp payoff/shr for e shares: Pr(Ai) (E[Pi, tfinal] -pi) + (1 -Pr(Ai)) (-pi) • Assume: E[Pi, tfinal] = Pi Pr(Ai) (Pi-pi) + (1 -Pr(Ai)) (-pi) !
Research Labs Market probability • Market probability MPr(Ai) • Probability at which the expected value of buying a share of Ai is zero • “Market’s” opinion of the probability • MPr(Ai) = pi/ Pi
Research Labs Price function derivation • Not unique; assume constraint, e. g. : pi/pj = Mi/Mj • MPr(Ai) = Mi. Si/W • pi = d. Mi/d. Si = Mi. T/W • Given current state (IC), number of shares received for m add. dollars is: sharesi(m)= m. Si/T - W-i/T-i(m/T+(T+m)/T-i Log((T+m)Mi/T/(Mi+m)))
Research Labs Price function derivation • Cost of buying n shares is costi(n) = sharesi-1(n) • No closed form; use e. g. Newton’s method
Research Labs Buying a set of outcomes • Let Q be a set of outcomes • Key simplifications • d. Si = d. Sj • d. Mi/d. Mj = const • Buying a set of outcome Q behaves like buying a single outcome with • MQ= i QMi • SQ = i QSi Mi/ j QMj
Research Labs Combinatorial market • Outcomes are base states • Events are sets of outcomes • 2 k possible events arising from k states • Use previous derivation to allow buying/selling arbitrary events
Research Labs Selling • A key advantage of DPM is the ability to cash out to lock gains / limit losses • “All money” case • Traders simply sell back to the market maker: double sided liquidity
Research Labs Selling • “Losing money” case: Each share is different. Composed of: 1. Original price refunded pri I(A) where I(A) is indicator fn 2. Payoff Pay I(A) • Selling do-able, more complicated; complexity can be hidden from traders to a degree
Research Labs Other price functions Share type Constraint/ Assumption Result Losing money p 1= P 2 p 2= P 1 Closed form cost() & shares() Losing money pi/pj = Mi/Mj Closed form cost() & shares() All money pi/pj = Mi/Mj Closed form shares() ; Numeric cost() All money pi/pj = Si/Sj Closed form cost() & shares()
Research Labs Initialization • Price functions are indeterminate when Mi=0 or Si=0 • Need to “seed” the market with money, shares per outcome; could come from • Patron • Ante • Capital - transaction fees • Acts like “b” in MSR Higher seed more risk, more initial liquidity • Unlike MSR, liquidity increases over time as shares are purchased
Research Labs Intermediate redistributions • For tracking repeated statistic • Interest rate • Real estate index • Oil prices • Redistribute money according to statistic at repeated intervals • pi/pj = Mi/Mj • No loss of money; continuity • Traditional MM (incl. MSR): requires additional subsidy
Research Labs Market scoring rule • Hanson 2002, 2003 • Special case of market maker: Automated, bounded loss • Market maker always stands willing to accept an (infinitesimal) trade at current prices • Full cost for some quantity is the integral over instantaneous prices • One example: prii(n) = e(n -a )/b cost(n) = j e(n -a )/b i i j j
Research Labs Market scoring rule • Market maker’s loss is bounded by b • Higher b more risk, more “liquidity” • Level of liquidity (b) never changes as wagers are made • Could charge transaction fee, put back into b (Todd Proebsting) • Much more to MSR: sequential shared scoring rule, combinatorial MM “for free”, . . . see Hanson 2002, 2003
Research Labs Mechanism comparison CDA Bounded liquidity dynamic info risk aggreg. X payoff vector fixed Increasing MM liquidity N/A CDAw MM X X MSR X PM X X N/A DPM X
Research Labs Future work • DPM call market • Combinatorial DPM implementation • Empirical testing What dist rule & price fn are “best”? • Real-valued (0 -infinity) outcomes
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