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### A Dynamic Pari-Mutuel Market for Hedging, Wagering, and Information Aggregation

Y!RL Spot Workshop onNew Markets, New Economics

- Welcome!
- Specific examples of new trends in economics, new types of markets
- virtual currency
- prediction (“idea”) markets
- experimental economics
- Interactive, informal
- ask questions
- rountable discussion wrap-up

Distinguished guests (thanks!)

- Edward CastronovaProf. Economics, Cal State Fullerton
- John LedyardProf. Econ & Social Sciences, CalTech
- Justin WolfersProf. Economics, Stanford

Schedule

11am-noon Castronova on the Future of Cyberspace Economies

noon-1pm Lunch provided

1pm-2pm Ledyard on ~ Information Markets and Experimental Economics

2pm-3pm Wolfers on ~ Prediction Markets, Play Money, & Gambling

3pm-3:30pm Pennock on Dynamic Pari-Mutuel Market for Hedging, Speculating

3:30pm-4pm Roundtable Discussion

David M. Pennock

paper to appear EC’04, New York

Economic mechanisms for speculating, hedging

- Financial
- Continuous Double Auction (CDA)stocks, options, futures, etc
- CDA with market maker (CDAwMM)
- Gambling
- Pari-mutuel market (PM)horse racing, jai alai
- Bookmaker (essentially like CDAwMM)
- Socially distinct, logically the same
- Increasing crossover

Take home message

- A dynamic pari-mutuel market (DPM)
- New financial mech for speculating on or hedging against an uncertain event; Cross btw PM & CDA
- Only mech (to my knowledge) to
- involve zero risk to market institution
- have infinite (buy-in) liquidity
- continuously incorporate new info;allow cash-out to lock in gain, limit loss

Outline

- Background
- Financial “prediction” markets
- Pari-mutuel markets
- Comparing mechs:PM, CDA, CDAwMM, MSR
- Dynamic pari-mutuel mechanism
- Basic idea
- Three specific variations; Aftermarkets
- Open questions/problems

= 6 ?

= 6

I am entitled to:

$1 if

$0 if

What is a financial“prediction market”?- Take a random variable, e.g.
- Turn it into a financial instrument payoff = realized value of variable

2004 CAEarthquake?

US’04Pres =Bush?

Real-time forecasts

- price expectation of random variable(in theory, in lab, in practice, ...huge literature)
- Dynamic information aggregation
- incentive to act on info immediately
- efficient market today’s price incorporates all historical information; best estimator
- Can cash out before event outcome
- BUT, requires bi-lateral agreement

Allocate risk (“hedge”)

insured transfers risk to insurer, for $$

farmer transfers risk to futures speculators

put option buyer hedges against stock drop; seller assumes risk

Aggregate information

price of insurance prob of catastrophe

OJ futures prices yield weather forecasts

prices of options encode prob dists over stock movements

market-driven lines are unbiased estimates of outcomes

IEM political forecasts

The flip-side of prediction: HedgingE.g. options, futures, insurance, ...Continuous double auctionCDA

- k-double auction repeated continuously
- buyers and sellers continually place offers
- as soon as a buy offer a sell offer, a transaction occurs
- At any given time, there is no overlap btw highest buy offer & lowest sell offer

http://us.newsfutures.com/

CDA with market maker

- Same as CDA, but with an extremely active, high volume trader (often institutionally affiliated) who is nearly always willing to sell at some price p and buy at price q p
- Market maker essentially sets prices; others take it or leave it
- While standard auctioneer takes no risk of its own, market maker takes on considerable risk, has potential for considerable reward

http://www.hsx.com/

Bookmaker

- Common in sports betting, e.g. Las Vegas
- Bookmaker is like a market maker in a CDA
- Bookmaker sets “money line”, or the amount you have to risk to win $100 (favorites), or the amount you win by risking $100 (underdogs)
- Bookmaker makes adjustments considering amount bet on each side &/or subjective prob’s
- Alternative: bookmaker sets “game line”, or number of points the favored team has to win the game by in order for a bet on the favorite to win; line is set such that the bet is roughly a 50/50 proposition

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

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

Dynamic pari-mutuel marketBasic idea

- Standard PM: Every $1 bet is the same
- DPM: Value of each $1 bet varies depending on the status of wagering at the time of the bet
- Encode dynamic value with a price
- price is $ to buy 1 share of payoff
- price of A is lower when less is bet on A
- as shares are bought, price rises; price is for an infinitesimal share; cost is integral

Dynamic pari-mutuel marketExample Interface

A

B

A

B

- Outcomes: A B
- Current payoff/shr: $5.20 $0.97

$3.27

$3.27

$3.27

$3.27

$3.27

$3.27

$3.25

sell 100@

$0.85

sell 100@

market maker

traders

sell 100@

sell 100@

$3.00

$0.75

$1.50

$0.50

sell 35@

sell 3@

$1.25

$0.25

buy 4@

buy 200@

buy 52@

$1.00

Dynamic pari-mutuel marketSetup & Notation

A

B

A

B

- Two outcomes: A B
- Price per share: pri1 pri2
- Payoff per share: Pay1 Pay2
- Money wagered: Mon1 Mon2 (Tot=Mon1+Mon2)
- # shares bought: Num1 Num2

How are prices set?

- A price function pri(n) gives the instantaneous price of an infinitesimal additional share beyond the nth
- Cost of buying n shares:
- Different assumptions lead to different price functions, each reasonable

Redistribution rule

- Two alternatives
- Losing money redistributed. Winners get: original money refunded + equal share of losers’ money
- All money redistributed. Winners get equal share of all money
- For standard PM, they’re equivalent
- For DPM, they’re significantly different

Losing money redistributed

- Payoffs: Pay1=Mon2/Num1 Pay2=.
- Trader’s exp pay/shr for e shares: Pr(A) E[Pay1|A] + (1-Pr(A)) (-pri1)
- Assume: E[Pay1|A]=Pay1 Pr(A) Pay1 + (1-Pr(A)) (-pri1)

Market probability

- Market probability MPr(A)
- Probability at which the expected value of buying a share of A is zero
- “Market’s” opinion of the probability
- MPr(A) = pri1 / (pri1 + Pay1)

Price function I

- Suppose: pri1 = Pay2 pri2=Pay1natural, reasonable, reduces dimens., supports random walk hypothesis
- Implies

MPr(A) = Mon1 Num1 Mon1 Num1 + Mon2 Num2

Deriving the price function

- Solve the differential equationdm/dn = pri1(n) = Pay2 = (Mon1+m)/Num2where m is dollars spent on n shares
- cost1(n) = m(n) = Mon1[en/Num2-1]
- pri1(n) = dm/dn = Mon1/Num2 en/Num2

Interface issues

- In practice, traders may find costs as the sol. to an integral cumbersome
- Market maker can place a series of discrete ask orders on the queue, e.g.
- sell 100 @ cost(100)/100
- sell 100 @ [cost(200)-cost(100)]/100
- sell 100 @ [cost(300)-cost(200)]/100
- ...

Price function II

- Suppose: pri1/pri2 = Mon1/Mon2also natural, reasonable
- Implies

MPr(A) = Mon1 Num1 Mon1 Num1 + Mon2 Num2

Deriving the price function

- First solve for instantaneous pricepri1=Mon1/Num1 Num2
- Solve the differential equationdm/dn = pri1(n) = Mon1+m(Num1+n)Num2

cost1(n) = m =

pri1(n) = dm/dn =

All money redistributed

- Payoffs: Pay1=Tot/Num1 Pay2=.
- Trader’s expected pay/shr for e shares:Pr(A) (Pay1-pri1) + (1-Pr(A)) (-pri1)
- Market probabilityMPr(A) = pri1 / Pay1

Price function III

- Suppose: pri1/pri2 = Mon1/Mon2
- Implies
- MPr(A) = Mon1 Num1 Mon1 Num1 + Mon2 Num2
- pri1(m) = cost1(m) =

Aftermarkets

- A key advantage of DPM is the ability to cash out to lock gains / limit losses
- Accomplished through aftermarkets
- All money redistributed: A share is a share is a share. Traders simply place ask orders on the same queue as the market maker

Aftermarkets

- Losing money redistributed: Each share is different. Composed of:
- Original price refundedpriI(A)where I(A) is indicator fn
- PayoffPayI(A)

Aftermarkets

- Can sell two parts in two aftermarkets
- The two aftermarkets can be automatically bundled, hiding the complexity from traders
- New buyer buys priI(A)+PayI(A) for pri dollars
- Seller of priI(A) gets $ priMPr(A)
- Seller of PayI(A) gets $ pri(1-MPr(A))

Alternative “psuedo” aftermarket

- E.g. trader bought 1 share for $5
- Suppose price moves from $5 to $10
- Trader can sell 1/2 share for $5
- Retains 1/2 share w/ non-negative value, positive expected value
- Suppose price moves from $5 to $2
- Trader can sell share for $2
- Retains $3I(A) ; limits loss to $3 or $0

Running comparison

[Hanson 2002]

Pros & cons of DPMs generally

- Pros
- No risk to mechanism
- Infinite (buying) liquidity
- Dynamic pricing / information aggregationAbility to cash out
- Cons
- Payoff vector indeterminate at time of bet
- More complex interface, strategies
- One sided liquidity (though can “hedge-sell”)
- Untested

Open questions / problems

- Is E[Pay1|A]=Pay1 reasonable? Derivable from eff market assumptions?
- DPM call market
- Combinatorial DPM
- Empirical testingWhat dist rule & price fn are “best”?
- >2 discrete outcomes (trivial?)Real-valued outcomes

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