Finish what s been left... CS286r Fall 08 Finish what s been left... 1

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1 Finish what s been left... CS286r Fall 08 Finish what s been left... 1

2 Perfect Bayesian Equilibrium A strategy-belief pair, (σ, µ) is a perfect Bayesian equilibrium if (Beliefs) At every information set of player i, the player has beliefs about the node that he is located given that the information set is reached. (Sequential Rationality) At any information set of player i, the restriction of (σ, µ) to the continuation game must be a Bayesian Nash equilibrium. (On-the-path beliefs) The beliefs for any on-the-equilibrium-path information set must be derived from the strategy profile using Bayes Rule. (Off-the-path beliefs) The beliefs at any off-the-equilibrium-path information set must be determined from the strategy profile according to Bayes Rule whenever possible. CS286r Fall 08 Finish what s been left... 2

3 A Perfect Bayesian Equilibrium Nature λ Bright Dull 1 λ Worker Worker C B C B Employer Employer λ β 1 λ 1 β H R H R H R H R (2, 2) (-1, 0) (4,-1) (1, 0) (2, 1) (-1, 0) (4, -2) (1, 0) β [0, 1] CS286r Fall 08 Finish what s been left... 3

4 Summary of Solution Concepts NE SPNE BNE PBE SE On-the-path beliefs On-the-path Off-the-path best response beliefs * * * Off-the-path best response * Beliefs derived from strategies according to Bayes rule Beliefs derived from strategies according to Bayes rule whenever possible There is a sequence of strategy-belief pairs that converge to the equilibrium strategy-belief pair. Each belief system is derived from the corresponding completely mixed strategy profile using Bayes rule. CS286r Fall 08 Finish what s been left... 4

5 Proper Scoring Rules, Theory of Security Markets, and Market Mechanisms October 1, 2008 CS286r Fall 08 Proper Scoring Rules, Theory of Security Markets, and Market Mechanisms 5

6 The Context We want to elicit and aggregate information about some uncertain events. CS286r Fall 08 Proper Scoring Rules, Theory of Security Markets, and Market Mechanisms 6

7 Ask An Expert An uncertain event (eg. tomorrow s weather) A forecaster has information about the event P( ) =0.8 P( ) =0.2 Ask the forecaster to report a probability distribution for the event How can we incentivize the expert to report and report truthfully? CS286r Fall 08 Proper Scoring Rules, Theory of Security Markets, and Market Mechanisms 7

8 How to incentivize experts? Pay the expert (in the future) based on his probability report and the actual outcome. First try: Pay the expert the probability that he reports for the true outcome P( ) =0.8 P( ) =0.2 The expert gets 0.8 if it is sunny, and 0.2 if it rains. If the expert tries to maximize his expected payoff max q p s q + (1 p s ) (1 q) = (1 p s ) + (2p s 1)q q = 1 if p s 1/2, and q = 0 if p s < 1/2. The expert now has the incentive to report, but not truthfully. CS286r Fall 08 Proper Scoring Rules, Theory of Security Markets, and Market Mechanisms 8

9 How to incentivize experts? Second try: Pay the expert the logarithmic of the probability that he reports for the true outcome + 5. P( ) =0.8 P( ) =0.2 The expert gets (5 + log 0.8) if it is sunny, and (5 + log 0.2) if it rains. If the expert tries to maximize his expected payoff max q p s (5 + log q) + (1 p s )(5 + log(1 q)) q = p s. The expert now reports truthfully! CS286r Fall 08 Proper Scoring Rules, Theory of Security Markets, and Market Mechanisms 9

10 Proper Scoring Rules An uncertain event with n mutually exclusive and exhaustive outcomes Report a probability estimate: r = (r 1, r 2,..., r n ) A scoring rule is a set of scoring functions {s 1 ( r), s 2 ( r),..., s n ( r)}, where s i ( r) is the payment when outcome i happens. A scoring rule is proper iif a risk-neutral agent maximizes his expected payoff by reporting truthfully. arg max r p i s i ( r) = p, i where p = {p 1, p 2,..., p n } is the true probability estimate of the agent. CS286r Fall 08 Proper Scoring Rules, Theory of Security Markets, and Market Mechanisms 10

11 Some Proper Scoring Rules Logarithmic Scoring Rule Quadratic Scoring Rule s i ( r) = a + b log(r i ) (b > 0) s i ( r) = a + 2br i b j r 2 j (b > 0) For binary event, s i ( r) = a b(1 r i ) 2 CS286r Fall 08 Proper Scoring Rules, Theory of Security Markets, and Market Mechanisms 11

12 Characterizations of Proper Scoring Rules Any positive linear transformation of a proper scoring rule is the same proper scoring rule. (McCarthy 56, Hendrickson and Buehlera 71) Any homogeneous and convex expected score function defines a proper scoring rule. Expected score function: H( r) = i r is i ( r) There are also proper scoring rules for eliciting probability distributions for continuous random variables [Matheson and Winkler 76, ] and for eliciting properties of probability distributions such as mean and quantiles [Savage 73, Cervera and J. Munoz 96, Gneiting and Raftery 07, Lambert et. al. 08]. CS286r Fall 08 Proper Scoring Rules, Theory of Security Markets, and Market Mechanisms 12

13 Combining Probability Distributions Need to combine m individual probability distributions into one consensus probability distribution T ( r 1, r 2,..., r m ) Some desired properties of the combined distribution External Bayesianity Performing Bayesian updates on individual distributions and then combining them = Performing Bayesian updates on the combined distribution Likelihood Property The probability of an outcome in the combined distribution only depends on the probabilities of the outcome in individual distributions. (Genest 84) T ( r 1, r 2,..., r m ) = r i if the above two properties are simultaneously satisfied. Dictator! CS286r Fall 08 Proper Scoring Rules, Theory of Security Markets, and Market Mechanisms 13

14 Utility of Money and Risk Attitudes u(x), where x is $ Risk attitudes Risk averse: u(x) is concave, eg. u(x) = log x Risk neutral: u(x) is linear, eg. u(x) = x Risk seeking: u(x) is convex, eg. u(x) = x 2 Absolute risk aversion: u (x)/u (x) CS286r Fall 08 Proper Scoring Rules, Theory of Security Markets, and Market Mechanisms 14

15 Risk Attitude and Hedging I m risk averse, u(x) = log x, and insurance company A is risk neutral u(x) = x I believe that my car might be stolen with probability ω 1 : car stolen ω 2 : car not stolen Expected Utility u(ω 1 ) = log(10, 000) u(ω 1 ) = log(20, 000) E(u) = I buy $10,000 insurance for $125. u(ω 1 ) = log(19, 875) u(ω 1 ) = log(19, 875) E(u) = Insurance company A also believes that P(car stolen) = 0.01 u(ω 1 ) = 9, 875 u(ω 1 ) = 125 E(u) = 25 > 0 I am happy to buy insurance. Insurancecompany A is happy to sell it. The transaction allocates risk. CS286r Fall 08 Proper Scoring Rules, Theory of Security Markets, and Market Mechanisms 15

16 Probability and Speculation Suppose that Im also risk neutral, u(x) = x. But I think that the probability for my car being stolen is much higher than 0.01, say 0.1. A $10,000 car insurance is worth.1(10, 000) +.9(0) = $1, 000 to me, but the insurance company only asks for $125. Too cheap! Buy the insurance, and I get $825 on expectation. I am speculating the insurance company. CS286r Fall 08 Proper Scoring Rules, Theory of Security Markets, and Market Mechanisms 16

17 Security Markets Note, the car insurance in fact is a contract $10,000 if car stolen, $0 otherwise Security markets generalize this to arbitrary states more than two parties What is traded: securities that specify state-contingent returns Terms of trade: price Market mechanism to allocate risk and allow speculation among participants. Speculation Information Aggregation CS286r Fall 08 Proper Scoring Rules, Theory of Security Markets, and Market Mechanisms 17

18 General Equilibrium General (competitive, Walrasian) equilibrium describes a simultaneous equilibrium of interconnected markets Definition: A price vector and allocation such that all agents making optimal demand decisions (positive demand = buy; negative demand = sell) all markets have zero aggregated demand (buy volume equals sell volume) CS286r Fall 08 Proper Scoring Rules, Theory of Security Markets, and Market Mechanisms 18

19 Rational Expectations Equilibrium [Grossman 81; Lucas 72] Fully Revealing Rational Expectations Equilibrium At a fully revealing rational expectations equilibrium, the equilibrium price reveals all private information. Agents behave as if they know the pooled information of all agents. CS286r Fall 08 Proper Scoring Rules, Theory of Security Markets, and Market Mechanisms 19

20 Common Criticism of REE How can REE be reached? CS286r Fall 08 Proper Scoring Rules, Theory of Security Markets, and Market Mechanisms 20

21 Can we agree to disagree? [Auman 76; Mckelvey 86; Mckelvey 90; Nielsen 90; Hanson 98] Procedural explanation: agents learn from prices Bayesian agents Agents begin with common priors, different private information Observe sufficient summary statistic (e.g. price) Update beliefs Converge to common posteriors CS286r Fall 08 Proper Scoring Rules, Theory of Security Markets, and Market Mechanisms 21

22 No-Trade Theorem [Milgrom and Stokey 82] Why trade? These markets are zero-sum games (negative sum w/ transaction fees) For all money earned, there is an equal (greater) amount lost; am I smarter than average? Rational risk-neutral traders will never trade Informally: Only those smarter than average should trade But once below average traders leave, average goes up Ad infinitum until no one is left Or: If a rational trader is willing to trade with me, he or she must know something I dont know CS286r Fall 08 Proper Scoring Rules, Theory of Security Markets, and Market Mechanisms 22

23 But People Do Trade Volume in financial markets, gambling is high Why do people trade? Different risk attitudes (insurance, hedging) Cant explain all volume Irrational (bounded rational) behavior Rationality arguments require unrealistic computational abilities, including infinite precision Bayesian updating, infinite game-theoretic recursive reasoning More than 1/2 of people think they are smarter than average Biased beliefs, inexperience, mistakes, etc. Note that its rational to trade as long as some participants are irrational CS286r Fall 08 Proper Scoring Rules, Theory of Security Markets, and Market Mechanisms 23

24 Some Market Mechanisms Call market Continuous double auction Pari-mutuel market Bookmaker CS286r Fall 08 Proper Scoring Rules, Theory of Security Markets, and Market Mechanisms 24

25 Call Market Stock market mechanism before 1800 Batch order processing Orders are collected over a period of time; Collected orders are matched at end of period Price is set such that demand=supply lim period 0: Continuous double auction CS286r Fall 08 Proper Scoring Rules, Theory of Security Markets, and Market Mechanisms 25

26 Call Market ; CS286r Fall 08 Proper Scoring Rules, Theory of Security Markets, and Market Mechanisms 26

27 Continuous Double Auction Call market repeated continuously Stock market mechanism Buy and sell orders continuously come in As soon as bid ask, a transaction occurs At any given time, there is a bid-ask spread Sometimes, CDA with a market maker A market maker is an extremely active, high volume trader (often institutionally affiliated) who is nearly always willing to buy at some price p and sell at some price q p CS286r Fall 08 Proper Scoring Rules, Theory of Security Markets, and Market Mechanisms 27

28 Pari-Mutuel Market Horse racetrack style wagering [Source: Pennock 04] CS286r Fall 08 Proper Scoring Rules, Theory of Security Markets, and Market Mechanisms 28

29 Pari-Mutuel Market Horse racetrack style wagering [Source: Pennock 04] CS286r Fall 08 Proper Scoring Rules, Theory of Security Markets, and Market Mechanisms 28

30 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 and/or subjective probs 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 CS286r Fall 08 Proper Scoring Rules, Theory of Security Markets, and Market Mechanisms 29