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A Unified Framework for Dynamic Pari-mutuel Information Market

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A Unified Framework for Dynamic Pari-mutuel Information Market. Yinyu Ye Stanford University Joint work with Agrawal (CS), Delage (EE), Peters (MS&E), and Wang (MS&E). Outline. Information Market Pari-mutuel Information Market LP Pari-mutuel Mechanism Dynamic Pari-mutuel Mechanism - PowerPoint PPT Presentation

Text of A Unified Framework for Dynamic Pari-mutuel Information Market

  • A Unified Framework for Dynamic Pari-mutuel Information MarketYinyu YeStanford UniversityJoint work with Agrawal (CS), Delage (EE), Peters (MS&E), and Wang (MS&E)

  • OutlineInformation MarketPari-mutuel Information MarketLP Pari-mutuel MechanismDynamic Pari-mutuel MechanismSequential Convex Pari-mutuel MechanismDesired Properties of SCPM and New Mechanism DesignExtensions to General Trading Market

  • What is Information MarketA place where information is aggregated via market for the primary purpose of forecasting events. Why:Wisdom of the Crowds: Under the right conditions groups can be remarkably intelligent and possibly smarter than the smartest person. James SurowieckiEfficient Market Hypothesis: financial markets are informationally efficient, prices reflect all known information

  • Sport Betting MarketMarket for Betting the World Cup WinnerAssume 5 teams have a chance to win the World Cup: Argentina, Brazil, Italy, Germany and France

  • Options for the MarketDouble Auction: Let participants trade directly with one anotherRequires participants to find someone to take the other side of their order (i.e.: the complement of the set of teams which they have selected)Appropriate method for markets with small number of states and large number of participantsCentralized Market MakerIntroduce a market maker who will accept or reject orders that he receives from market participantsMarket organizer may be exposed to some risk This approach works better in thinly traded marketsGreater liquidity can be induced by allowing multi-lateral order matchingLower transaction costs (no search costs for the participants)Problem: How should the market organizer fill orders in such a manner that he is not exposed to any financial risk?

  • Central Organization of the MarketBelief-basedCentral organizer will determine prices for each state based on his beliefs of their likelihoodThis is similar to the manner in which fixed odds bookmakers operate in the betting worldGenerally not self-fundingPari-mutuelA self-funding technique popular in horseracing betting.

  • Pari-mutuel Market Model IDefinitionEtymology: French pari-mutuel, literally, mutual stake A system of betting on races whereby the winners divide the total amount bet in proportion to the sums they have wagered individually (after deducting management expenses).Example: Parimutuel Horseracing Betting

  • World Cup Betting MarketMarket for World Cup WinnerWed like to have a standard payout of $1 per share if a participant has a winning order.Combinatorial Orders

  • Pari-mutuel Market Model IICombinatorial Betting Language in the MarketN possible states of the world (one will be realized)n participants who, trader k, submit orders to a market maker containing the following information:ai,k - state bid (either 1 or 0)k bid price per sharelk limit on share quantityMarket maker will determine the following:xk order fill/# of awarded sharespi state price/beliefs/probabilitiesCall or dynamic auction mechanism is used.If an order is accepted and correct state is realized, market maker will pay the winning order a fixed amount $1 per share.

  • Research Evolution

  • LP Pari-mutuel Market MechanismBoosaerts et al. [2001], Lange and Economides [2001], Fortnow et al. [2003], Yang and Ng [2003], Peters et al. [2005], etcAn LP pricing mechanism for the call auction market

  • World Cup Betting ResultsOrders Filled

    State Prices

  • Other IssuesHow to make state prices uniqueHow to create initial funding to the marketHow to incorporate the market makers own belief into the market

    Non-convex formulation with unique state prices/beliefs by Lange and Economides [2005]

  • Belief-Based and Risk NeutralThis mechanism has a fixed price i for all i

  • Belief-Based and Risk Averseb is a combination weight factor in [0 1]

  • Convex Pari-mutuel Market MechanismPeters et al. [2005], etcTheorem (Peters et al. 2005) Convex programming of call auction has unique state prices p(b) that are identical to those of the non-convex formulation of Lange and Economides (2005).

  • Utility-Maximization InterpretationLet the concave and increasing utility function be ln(.), and i be the market makers probability belief on state i. Then, the objective of the market maker is the worst case profit combined with an expected utility value of the contingent state realization. Here, b is a positive combination weight factor:

  • Dynamic Pari-mutuel Market ModelTraders come one by one with order (a, , l)Market maker has to make an order-fill decision as soon as an order arrivesmay need to accept bets that do not have a matching bet yet.Market maker still hopes to pay the winners almost completely from the stakes of losersto update state prices reflect the traders' aggregated belief on outcome states

  • Desirable Properties of MechanismsEfficient computation for price updateTruthfulness (in myopic sense) Bidding true value of a bet should be dominant strategy for each trader (if he or she is a one-time trader)PropernessA dominant strategy for traders is to place bets on outcome states so that resulting price reflects his or her true beliefstronger condition than truthfulnessBounded worst-case lossNet amount the market maker may have to pay the winners at the endRisk attitude of the market-makerMarket organizer takes certain risk when accepting bets that are not matched by the current bets in the systemThe risk attitude of market maker determines the dynamics of marketextreme risk averseness implies that no bet will ever get accepted.

  • Background: Existing Mechanisms IMarket Scoring RuleTraders report their beliefs/prices, p, on outcome states directlyPayment is determined by a scoring rule, si( p ), on reported price vector p in the probability simplex S={ p 0: pi=1 }For some positive constant b:Logarithmic Market Scoring Rule (LMSR) Hanson [2003]

    Quadratic Market Scoring Rule (QMSR)

  • Market Scoring RuleSuppose constant b=0.1 and you bet the distribution p=(0.2, 0.3, 0.2, 0.25, 0.05) on the five teams. Then, if Brazil wins, your reward for each share under (LMSR) is 0.1ln(.3) + 1 = .87

  • Background: Existing Mechanisms IICost-Function Based Scoring Rule (Chen and Pennock 2007)Trader submits an order quantity characterized by the vector v, where vi represents the number of shares that the trader desires over state i The total fee charge to the trader

    where C( q ) is a cost function of the current outstanding share quantity vector q.Instantaneous price vector would be C( q ) reflecting aggregated beliefs/probabilities.

  • Background: Existing Mechanisms IIITheorem (Chen and Pennock 2007) Every scoring rule admits a cost-function representation, C(q), where

    LMSR:

    QMSR:

    Note that the quadratic cost scoring rule cannot guarantee the price/probability vector nonnegative

  • Background: Existing Mechanisms IVSequential Convex Pari-mutuel Mechanism (Peters et al. 2007) for an arrival order (a,, l )

    where q is the current outstanding share quantity vector, e is the vector of all ones, and x it the order fill variable. Prices are the optimal Lagrange multipliers of the convex optimization problem

  • Background: Existing Mechanisms VIt turns out that one can use the KKT optimality conditions to create a quick update scheme to solve the SCPM model for an arrival order, instead of needing to solve the full convex program each time.

    Theorem (Peters et al. 2007) The SCPM problem can be solved in double-logarithmic time, that is, log log(1/) arithmetic operations.

    The computational complexity of the three described mechanisms are essentially identical.

  • QuestionsWhat are the common features and differences among these mechanisms?Why some properties are satisfied or unsatisfied by a mechanism?What type of cost-functions imply a valid scoring rule?How to compare and rationalize different mechanismsIs there new and better mechanism yet to be discovered?

  • In this WorkA unified framework is developed that subsumes existing mechanismsestablishes necessary and sufficient conditions for satisfying certain desirable propertiesprovides a tool for designing new mechanisms with all desirable properties

  • Unified Pari-mutuel Market MechanismGeneralized Sequential Convex Pari-mutuel Mechanism for an arrival order (a,, l )

    where q is the current outstanding share quantity vector, e is the vector of all ones, x it the order fill variable, and u(s) is any (expected) concave and increasing value function of slack shares retained by the market maker.

  • Prices in SCPMMarket maker maximization principle: the unified framework is to balance market makers revenue from the arrival trader and (future) value

    Prices/beliefs are the optimal Lagrange multipliers of the convex optimization problem with maximizers (x*,s*,z*), and they are

  • The Main ResultsEvery scoring rule or cost function mechanism is the SCPM corresponding to a specific concave and increasing value function. Conversely, every concave and increasing value function in SCPM induces a scoring rule or cost function mechanism and can be truthfully implemented. The properties of the value function and its derivatives, such as boundedness, smoothness, span, etc, determine other desired or undesired properties of the mechanism, such as the worst-case loss, properness, risk-attitude, etc.

  • Value Functions of Existing MechanismsLMSR:

    QMSR*:

    Log-SCPM:

  • Other UtilitiesLinear-SCPM:

    Min-SCPM:

    Exp-