Decision Markets With Good Incentives

1. Decision Markets With Good Incentives YilingChen (Harvard), Ian Kash (Harvard), Internet and Network Economics, 2011. amitsomech@gmail.com

2. Prediction Markets • Markets used for predictionthe outcome of an event Project Manager ?

3. Decision Markets • Using (prediction) markets for decision making. • For example: Deciding between hiring Alice or Bob. Project Manager ?

4. Decision Markets • Decision maker creates two conditional prediction markets: #1: Will we complete testing on time ?| Alice is hired ---0.66 #2: Will we complete testing on time ?| Bob is hired --- 0.44 Project Manager ? • 0.44 • 0.66

5. Decision Markets • DM considers the final prediction (0.44,0.66), then chooses action according to a decision rule : • For example: MAX Decision Rule – choose the Actionwith greater probability to achieve the desired outcome Project Manager ? • 0.44 • 0.66

6. Decision Markets • DM waits for the outcome. • DM pays the experts according to: • Final prediction (0.44,0.66) • Action (Hiring Alice) • Outcome (Testing completed on time ) Testing completed on time Testing delayed project DD

7. Decision Market - Definition • Prediction market is a special case of decision market. • Both use the same sequential market structure. • Decision market uses a decision rule to pick from a set of actions before the outcome is observed. • Which action is chosen may affect the likelihood an outcome occurs. ? • 0.44 • 0.66 Testing completed on time Sequential Market yields final prediction Decision Maker chooses an action An outcome occurs Scoring the experts

8. Outline • What are Decision Markets • explanation • Model: notations and definitions • Problem with myopic incentives • Incentive in Decision Markets • Decision Scoring rules • Existence of a strictly proper decision market • Necessity of full support in decision scoring rules • Optimal Decision Markets Suggestions

9. Model: Assumptions • About experts and the market: • Experts can only observe prior predictions before making their own. • After the market ends, a final, consensus prediction is made. • Experts are utility driven – no extern incentives. • About Decision making: • Decision maker chooses only one action. • *Decision maker can draw an action stochastically. • The method of decision can be described as a function

10. Model: Notations and Definitions From prediction markets: • O – set of possible outcomes. {finished on time, did not finish on time} • ∆(O) – set of probability distribution over outcomes. • pt∆(O) –prediction made at round t. • Scoring Rule: A function for scoring a prediction p ∆(O) ,according to outcome o* O . • a shorthand:

11. Model: Notations and Definitions (2) For Decision Market:new! • A - finite set of actions {Hiring Alice, Hiring Bob} • ∆(O) |A | - set of conditional distributions, one for each action. • Each expert predicts outcome for each and every action. • The market is being held simultaneously for all actions. • Pt∆(O) |A | – prediction made at round t (for all actions). • ∆(O) |A | - final report.

12. Model: Notations and Definitions (3) • Decision Rule:A function • D() - Applied to the final report • ∆(A) – is a set of distributions: drawing an action a* from A • Shorthands: • d – the distribution over all actions • da– the likelihood action a is drawn from the set A • Examples: • MAX: Note that D() is a distribution. We will show that it is necessary for creating myopic incentive compatibility.

13. Decision Market Model 1) The market opens. • P0 ∆(O) |A| – Initial Prediction in the market. • Pt ∆(O) |A| –Prediction at round t. 2) The market closes at round , last prediction is . 3) Decision maker applies the decision rule: D( 4) Decision maker draws a single action a* according to d. 5) The outcome o* is revealed. 6) Decision maker pays the experts. How?

14. Outline • What are Decision Markets • explanation • Model: notations and definitions • Problem with myopic incentives • Incentive in Decision Markets • Decision Scoring rules • Existence of a strictly proper decision market • Necessity of full support in decision scoring rules • Optimal Decision Markets Suggestions

15. Decision Market Model 1) The market opens. • P0 ∆(O) |A| – Initial Prediction in the market. • Pt ∆(O) |A| –Prediction at round t. 2) The market closes at round , last prediction is . 3) Decision maker applies the decision rule: D( 4) Decision maker draws a single action a* according to d. 5) The outcome o* is revealed. 6) Decision maker pays the experts. How? Apply a scoring rulefor the selected action

16. So, What Is the Problem? Consider the following scenario: • Decision maker creates a Decision market for choosing Alice or Bob. • Decision rule: MAX (i.e., market maker hires the candidate with better predicted probability) • Payment method: experts are paid after the candidate is hired, and the outcome is revealed , according to the scoring rule. ? • 0.44 • 0.66 Testing completed on time Sequential Market yields final prediction Decision Maker chooses an action An outcome occurs Scoring the experts

17. So, What Is the Problem? (2) • Current Market values at some round t: • Alice: 0.2 • Bob: 0.8 • An expertwith belief (Alice: 0.75,Bob: 0.8) enters the market. • What will be the expert’s prediction? • (Alice:0.75,Bob:0.8) raise Alice’s market value to 0.75. • (Alice:0.81,Bob:0.8) Raise Alice’s market value to 0.81. • (Alice:0.75,Bob:0.74) Lower Bob’s market value to 0.74 and raise Alice’s to 0.75

18. So, What Is the Problem? (2) • Current Market values: • Alice: 0.2 • Bob: 0.8 • An expertwith belief (Alice: 0.75,Bob: 0.8) enters the market. • What will be the expert’s prediction? • raise Alice’s market value to 0.75. • Raise Alice’s market value to 0.81. • Lower Bob’s market value to 0.74 and raise Alice’s to 0.75. • Do not participate.

19. So, What Is the Problem? (3) A. Truthful reporting: • The expert raises Alice’s market value to 0.75 • Decision maker chooses Bob (has prob. 0.8) • Expert get nothing (he doesn’t own Bob shares) B. Overbuying Alice: • The expert raises Alice’s market value to 0.81 • Decision maker chooses Alice (has prob. 0.81) • Expert’s payment: • Raising from 0.2 to 0.75: Positive • Raising from 0.75 to 0.81: Negative • Overall: Positive

20. So, What Is the Problem? (4) C. Leveling Alice and Artificially Lowering Bob: • The expert raises Alice’s market value to 0.75 • The expert lowers Bob’s market value to 0.74 • Decision maker chooses Alice (has prob. 0.75) • Expert’s payment: • Raising from 0.2 to 0.75: Positive

21. So, What Is the Problem? (5) Is C better than B? Consider then 2nd expert (with the same belief [Alice:0.75,Bob:0.8]): • case C: • Market value is: Alice – 0.75, Bob- 0.74 • Expert #2 will raise Bob’s value back to 0.8! • caseB: • Market value is: Alice – 0.81, Bob- 0.8 • Expert #2: • Buying short on Alice will result in no payoff • Thus, Expert #2 do nothing!!

22. Outline • What are Decision Markets • explanation • Model: notations and definitions • Problem with myopic incentives • Incentive in Decision Markets • Decision Scoring rules • Existence of a strictly proper decision market • Necessity of full support in decision scoring rules • With Strictly properness, preferred action can be chosen W.P close to (but not) 1. • Optimal Decision Markets Suggestions

23. Scoring Experts: Decision Scoring Rule • Instead of scoring by a scoring rule ( ), with respect only to the outcome and the prediction for the chosen action, we use a decision scoring rule. • Decision scoring rule: • Written • Mapping an action, outcome, decision policyand predictionto the extended reals.

24. Decision Scoring Rule: Example • Decision Rule: d(P) • Decision Scoring rule: • so- is a logarithmic scoring rule :1+logx • So if Alice is hired, and final prediction is • Alice:0.25, Bob:0.75 • dAlice= 0.2, dBob=0.8 • SAlice,finished on time,=5*(1+log(0.25)) • SBob,finishedon time,=1.25*(1+log(0.75))

25. Decision Scoring Rule: • Expected score: • Q – the expert’s personal belief • P – the expert’s prediction This is the sum of possible scores weighted by how likely each score: to be realized • (Strictly) Properness: • For all beliefs Q, distributions d and d’ and prediction P • Strictly properness: the inequality is strict unless P=Q

26. Myopic Incentives in Prediction Vs. Decision Markets • da- porbability for choosing action a • Qa,o – (vector) belief of ouctomeo for each action a • Sa,o– Decision scoring rule with respect to the final prediction P and the probability vector d for choosing an action

27. Outline • What are Decision Markets • explanation • Model: notations and definitions • Problem with myopic incentives • Incentive in Decision Markets • Decision Scoring rules • Existence of a strictly proper decision market • Necessity of full support in decision scoring rules • With Strictly properness, preferred action can be chosen W.P close to (but not) 1. • Optimal Decision Markets Suggestions

28. Strictly Proper Decision Market Existence of a strictly proper decision market • Theorem 1: let D be a decision rule (with full support *). Then there exists a decision rule S such that (D,S) is strictly proper

29. Strictly Proper Decision Market (2) • Existence of a strictly proper decision market • Proof: for any strictly proper scoring rule s: Then the expected payment is: Prediction Market Scoring rule Linearity of Expectation

30. Strictly Proper Decision Market (3) Necessity of full-support • Full support decision rule: if

31. This Model is Still Not Optimal • We proved that MAX decision rule can not be used in myopic incentive compatible decision market • A stochastic decision rule with full support is crucial for obtaining myopic incentive compatibility • In practice, no decision maker will knowingly choose the wrong decision, even with small probability

32. Optimal Decision Markets • Right Action Rules (Chen[2012]) • Compensation function: (Boutilier [2012]) • Fool the agents (TA example)