DECISION MARKETS WITH GOOD INCENTIVES Yiling Chen (Harvard), Ian Kash (Harvard), Internet and Network Economics,

Prediction Markets Project Manager Markets used for prediction the outcome of an event ?

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

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

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 Action with greater probability to achieve the desired outcome Project Manager ?

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

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. Testing completed on time ? Sequential Market yields final prediction Decision Maker chooses an action An outcome occursScoring the experts

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

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

Model: Notations and Definitions

Model: Notations and Definitions (2)

Model: Notations and Definitions (3)

Decision Market Model

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

Decision Market Model Apply a scoring rule for the selected action

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. Testing completed on time ? Sequential Market yields final prediction Decision Maker chooses an action An outcome occursScoring the experts

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

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

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

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

So, What Is the Problem? (5) Is C better than B? Consider then 2 nd expert (with the same belief [Alice:0.75,Bob:0.8]): case C: Market value is: Alice – 0.75, Bob Expert #2 will raise Bob’s value back to 0.8! case B: 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!!

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

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 policy and prediction to the extended reals.

Decision Scoring Rule: Example

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 Decision Scoring Rule:

Myopic Incentives in Prediction Vs. Decision Markets Decision MarketsPrediction Markets Expected payment of a single expert (strictly*) Proper scoring rule *inequality is strict unless q=p  d a - porbability for choosing action a  Q a,o – (vector) belief of ouctome o for each action a  S a,o – Decision scoring rule with respect to the final prediction P and the probability vector d for choosing an action

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

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

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

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

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

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