A Utility Framework for Bounded-Loss Market Makers

1. 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

2. What will we see?

3. 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

4. 1 / 2Introduction - • 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.

5. 2 / 2Introduction - • 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

6. Background • Scoring rules: • 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:

7. 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

8. 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.

9. 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:

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

11. 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

12. 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

13. 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.

14. 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

15. 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).

16. 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:

17. Relationship Between MSR andUtility-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.

18. 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:

19. 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

20. 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.

21. 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:

22. 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.

23. 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.

24. 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.

25. 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:

26. 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:

27. Questions?

28. Thank you!