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Dynamic Pari-Mutuel Market Strategy Guide

Learn about Dynamic Pari-Mutuel Markets for hedging, wagering, and information aggregation. Discover how prices are set, market probabilities calculated, and more. Understand buying and selling strategies, combinatorial markets, and the advantage of cashing out. Explore ways to initialize the market and manage intermediate redistributions effectively. Dive into Market Scoring Rules for automated trading with bounded losses.

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Dynamic Pari-Mutuel Market Strategy Guide

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  1. A Dynamic Pari-Mutuel Market for Hedging, Wagering, and Information Aggregation David M. PennockMike Dooley Previous Version Appears in EC’04, New York

  2. A B What is a pari-mutuel market? • E.g. horse racetrack style wagering • Two outcomes: A B • Wagers:

  3. What is a pari-mutuel market? A B • E.g. horse racetrack style wagering • Two outcomes: A B • Wagers: 

  4. What is a pari-mutuel market? A B • E.g. horse racetrack style wagering • Two outcomes: A B • Wagers: 

  5. $ on B 8$ on A 4 1+ = 1+ =$3 total $ 12$ on A 4 = = $3 What is a pari-mutuel market? A B • E.g. horse racetrack style wagering • Two outcomes: A B • 2 equivalentways to considerpayment rule • refund + share of B • share of total 

  6. 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

  7. Dynamic pari-mutuel marketBasic idea 1 1 1 1 1 1 1 1 1 1 1 1

  8. Dynamic pari-mutuel marketBasic idea 1 1.1 1 1.3 0.2 0.4 1.6 0.9 2 2.5 3

  9. Dynamic pari-mutuel marketSetup & Notation A2 ... Ak A1 • Outcomes: Ai • Prices (per share): pi • Payoffs (per share): Pi • Money wagered on i: Mi • # shares purchased of i: Si • Total money: T = jMj • Share-weighted total: W = jSjMj • “Other” money: T-i = T - Mi • “Share-weighted other” $: W-i = W - SiMi

  10. What is a share? • Two alternatives; Share of • losing money onlyWinners get: original money refunded + equal share of losers’ money • all moneyWinners get equal share of all money • For standard PM, they’re equivalent • For DPM, they’re not

  11. 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: 0n pi(n) dn • Different reasonable assumptions lead to different price functions

  12. ! “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)

  13. 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

  14. Price function derivation • Not unique; assume constraint, e.g.: pi/pj = Mi/Mj •  MPr(Ai) = MiSi/W •  pi = dMi/dSi = MiT/W • Given current state (IC), number of shares received for m add. dollars is: sharesi(m)=mSi/T - W-i/T-i(m/T+(T+m)/T-iLog((T+m)Mi/T/(Mi+m)))

  15. Price function derivation • Cost of buying n shares is costi(n) = sharesi-1(n) • No closed form;use e.g. Newton’s method

  16. Buying a set of outcomes • Let Q be a set of outcomes • Key simplifications • dSi = dSj • dMi/dMj = const • Buying a set of outcome Q behaves like buying a single outcome with • MQ= iQMi • SQ = iQSi Mi/jQMj

  17. Combinatorial market • Outcomes are base states • Events are sets of outcomes • 2k possible events arising from k states • Use previous derivation to allow buying/selling arbitrary events

  18. 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

  19. Selling • “Losing money” case: Each share is different. Composed of: • Original price refundedpriI(A)where I(A) is indicator fn • PayoffPayI(A) • Selling do-able, more complicated; complexity can be hidden from traders to a degree

  20. Other price functions

  21. 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 MSRHigher seed  more risk, more initial liquidity • Unlike MSR, liquidity increases over time as shares are purchased

  22. 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

  23. 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(ni-ai)/b cost(n) = j e(nj-aj)/b

  24. 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

  25. Mechanism comparison

  26. Future work • DPM call market • Combinatorial DPM implementation • Empirical testingWhat dist rule & price fn are “best”? • Real-valued (0-infinity) outcomes

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