A Utility Framework for Bounded-Loss Market Makers Yiling Chen (Yahoo! Research), David M. Pennock (Yahoo! Research) 2007 Presentation by: Yonatan Herzig 21/11/12

What will we see?

Outline Introduction Background Utility-Based Market Makers Relationship Between MSR and Utility- Based Market Makers Cost-Function Formulation of Market Makers Liquidity and Market Maker Loss

1 / 2 Introduction - In prediction markets price discovery is the market designer’s end goal and trading is a means to that end. Liquidity is needed for price discovery. Prediction market are expected to be thinly traded, since meeting trader demand is not the market’s primary goal.

2 / 2 Introduction - An automated market maker can improve liquidity. The market maker continually announces prices offering both to buy and to sell some quantity of the security, adjusting his prices in programmatic response to trader demand. a prediction market designer may well subsidize a market maker that expects to lose some money. In return for improving trader incentives, liquidity, and price discovery. This paper presents a framework for Bounded-Loss Market Makers

Background Scoring rules: MSR (Hanson [2003, 2007] ): let v represent a discrete random variable with N mutually exclusive and exhaustive outcomes. let be a probability estimate for variable v. A scoring rule is a list of scoring functions, such that a score is assigned to if outcome i of the random variable v is realized. MSR (Hanson [2003, 2007] ): A patron subsidizes an automated market maker in order to improve liquidity. The patron loses no more than a fixed constant. The market maker begins by setting an initial probability estimate, . Every trader can change this probability to a new one if he is willing to pay the appropriate scoring rule. The scoring rule:

What is the utility of money? Utility-Based Market Makers What is the utility of money? The utility function, u(m), of money is a measure for the satisfaction associated with a particular level of wealth. The marginal utility, u’(m), depicts how much more satisfied an agent will be with gaining more wealth. It indicates the agent’s desire to take risks for gaining more – his risk aversion. Risk averse Risk indifferent Risk loving

Utility-Based Market Makers Risk neutral probability u(m) – utility function of money. - Subjective probability estimate for the forecast variable. - Wealth vector across outcome results. The agent’s risk neutral probabilities: Risk-neutral probabilities are the price levels that the agent is indifferent between buying or selling an infinitely small number of shares. Equal to subjective probability when agent is indifferent to risk.

Utility-based market makers The main idea: The market maker equates the instantaneous security prices to his current risk-neutral probabilities , and is always willing to accept infinitely small buy or sell orders of Arrow-Debreu contracts at these prices. A trader comes and buy an infinitely small number of security i, . Then, the market maker’s wealth in state increases by , and for state i, decreases by Let be the vector of the total number of outstanding shares. Then:

Utility-based market makers Thus, this equation system defines a utility based market maker:

Loss of utility based market makers At any time of the market, a utility-based market maker satisfies: where k is constant. Proof: Lets derivate by qi: Thus, the expected utility is constant during all process of trading. A market maker can choose initial wealth vector, , to start the market, and k equals

Loss of utility based market makers A market maker has bounded loss if and only if there exists a real constant l such that for any that meets , l ≤ mj is satisfied for all j. For a real-valued, continuous and strictly increasing utility function u(m), any fixed subjective probability estimate π whose elements are nonzeros, and a utility-based market maker who sets prices according to its rules: the necessary and sufficient condition for the market maker to have bounded loss for any feasible expected utility level k is that at least one of the following conditions is satisfied: The domain of u(m) is bounded below The range of u(m) is bounded above but not bounded below

Loss of utility based market makers Semi intuitive proof: The domain of u(m) is bounded below If the market maker plays by the defined rules, u(m) is always defined, so m is always bounded below. The range of u(m) is bounded above but not bounded below lets assume m is not bounded below, so at a certain point for some i: , then: But u(m) is bounded above: a contradiction.

Hyperbolic Absolute Risk Aversion (HARA) The hyperbolic absolute risk aversion (HARA) class of utility functions contains most popular parametric families of utilities. Generic form: Where M is a real number, γ is an extended real number, and α > 0. The utility function is defined on the domain : : family of constant relative risk aversion (CRRA) utility functions It can be verified that a utility-based market maker using a non- linear HARA utility function is guaranteed to have bounded loss

Weighted Pseudospherical Scoring Rules The scoring rule function of this class is (Jose et al. [2006] ): Where b>0 and the reported probability estimate is weighted by a base-line probability estimate . Weighted pseudospherical scoring rules are strictly proper for any real β. β → 1 : : β = 2 and π is uniform : the spherical scoring rule The quadratic scoring rule (2), however, does not belong to this class. A MSR market maker’s loss is bounded when using any one of these rules (the payment when probability 1 is reported for the true outcome is finite).

Expected-Utility-Maximization Problem show that the weighted pseudospherical scoring rules can arise from the solution of an expected-utility-maximization problem of a forecaster against a betting opponent (Jose et al. [2006] ). A market with a market maker can be viewed as a sequence of two- person betting, each happening between a trader (A) and the market maker (B). This kind of bet can be seen as selecting a vector , s.t. the wealth vectors for A and B are and , and they both accept according to their subjective probability estimates on v. In a utility based market, a (risk neutral) agent will change into f for the market maker such that the trader’s expected wealth is maximized: The market maker is willing to accept only if the expected utility stays constant. We get the next optimization problem:

Relationship Between MSR and Utility-Based Market Makers The Market Maker Equivalence Theorem: A utility-based market maker who has a subjective probability estimate and a HARA utility function with γ = 0 is equivalent to a market scoring rule market maker who utilizes a π-weighted pseudospherical scoring rule with β = 1− 1/γ . More specifically, for example: A negative exponential utility market maker (γ = ±∞) is equivalent to an MSR market maker with a weighted logarithmic scoring rule. If starting the market with uniform prior probabilities, a negative exponential utility market maker is equivalent to a logarithmic MSR market maker.

Cost-Function Formulation The market contains a total of N securities, each paying $1 per share if its corresponding outcome happens. q is the vector of all quantities of shares held by traders. Market utilizes a cost function, , that records the total amount of money traders have spent as a function of the total number of shares held of each security. Agent willing to change to , pays the market maker the going price of security i is: the sum of prices always equals 1 to ensure no arbitrage. Thus, these properties hold:

Cost Functions and Utility-Based Market Makers utility-based market maker has a cost function that is defined by: , where k is constant. since is the total money collected by the market maker and qj is the total money that the market maker needs to pay if outcome j happens. According to the implicit function theorem in mathematical analysis, we can see that if the utility function u(m) is continuous, differentiable, and strictly increasing, there exists a unique function

Cost Functions and Utility-Based Market Makers Actually what we are trying to solve is (assuming uniform ): For example, for the logarithmic utility function with b>0: , , the cost function for two outcome event is: Verify it for homework ;) And for the negative exponential utility function: The price functions can be calculated by differentiation.

Cost Functions and Market Scoring Rules For scoring rules, such as quadratic coring rule, that do not belong to the weighted pseudospherical scoring rule class, we describe how to make the translation directly In MSR, when a trader wants to change to gets profit Under the cost function formulation, A trader with a estimation, will buy or sell shares, until the market becomes . He changes the quantity to , and when outcome i happens, his profit is For MSR market maker, and cost function formulation to be equivalent, the trader should profit the same amount for all i:

Cost Functions and Market Scoring Rules Solving the last equation system for LMSR leads to: For the quadratic scoring rule, we get: Price functions can be obtained by differentiation.

What did we see? we have established equivalence translations for restricted classes. The cost function formulation seems most natural for implementation purposes. Market scoring rules make many analysis such as loss of market makers straightforward, however directly converting them to cost-function formulations is not always easy. Utility-based market makers, whose cost functions are easy to find, connect the MSR with cost-formulation for a large class of scoring rules.

Liquidity and Market Maker Loss To address liquidity in prediction markets (no bid-ask spread or market depth), we examine The instantaneous liquidity for security i at : The slope of the price function, ∂pi(q)/∂qi, approximates the bid-ask spread for the differentiable price function pi(q). It approximates the price difference between buying one share of security i from the market maker and selling one share to the market maker. Moreover, ∂pi(q)/∂qi inversely relates to the instantaneous depth of the market for security i. A smaller ∂pi(q)/∂qi means that it requires more shares to drive price up or down by one unit, hence implying a deeper market.

Loss and Instantaneous Liquidity We consider a class of market makers whose cost functions are symmetric and second-order differentiable (so price function is differentiable ). Symmetric cost functions: if is a permutation of , then Same goes for price functions: They have the same form for every i. This means that if all qi’s are equal, all pi’s are equal to 1/N. Lets assume and . a market maker’s worst-case loss is bounded by:

Loss and Instantaneous Liquidity Theorem: For all market makers who utilize a second-order differentiable and symmetric cost function over N outcomes and maintain an instantaneous liquidity level no lower than ρ for all , the minimum worst-case loss is Proof: Minimum loss is when liquidity is the lowest. So this happens when: Lets calculate the market loss as we’ve learned:

Questions?

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