DECISION MARKETS WITH GOOD INCENTIVES

Yiling Chen (Harvard), Ian Kash (Harvard), Internet and Network Economics,2011.

amitsomech@gmail.com

Prediction Markets

Project Manager

• Markets used for prediction the outcome of an event

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

Project Manager

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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

?0.660.44

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

?0.660.44

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 ? 0.660.44

Sequential Market yields final prediction

Decision Maker chooses an action

An outcome occurs Scoring the experts

OutlineWhat are Decision Markets

explanationModel: notations and definitionsProblem with myopic incentives

Incentive in Decision MarketsDecision Scoring rules Existence of a strictly proper decision marketNecessity 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 DefinitionsFrom 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:

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.

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.

Decision Market Model1) 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?

OutlineWhat are Decision Markets

explanationModel: notations and definitionsProblem with myopic incentives

Incentive in Decision MarketsDecision Scoring rules Existence of a strictly proper decision marketNecessity of full support in decision scoring rules

Optimal Decision Markets Suggestions

Decision Market Model1) 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 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 ? 0.660.44

Sequential Market yields final prediction

Decision Maker chooses an action

An outcome occurs Scoring 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 0.75.B. (Alice:0.81,Bob:0.8) Raise Alice’s market value to 0.81.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 0.75.B. Raise Alice’s market value to 0.81.C. Lower Bob’s market value to 0.74 and raise Alice’s to 0.75.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 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!

• 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!!

OutlineWhat are Decision Markets

explanationModel: notations and definitionsProblem with myopic incentives

Incentive in Decision MarketsDecision Scoring rules Existence of a strictly proper decision marketNecessity of full support in decision scoring rulesWith 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 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,finished on time,=1.25*(1+log(0.75))

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 Markets Prediction MarketsExpected payment of a single expert

(strictly*) Proper scoring rule

*inequality is strict unless q=p

da- porbability for choosing action a Qa,o – (vector) belief of ouctome o 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

OutlineWhat are Decision Markets

explanationModel: notations and definitionsProblem with myopic incentives

Incentive in Decision MarketsDecision Scoring rules Existence of a strictly proper decision marketNecessity of full support in decision scoring rulesWith Strictly properness, preferred action can be chosen W.P close

to (but not) 1.Optimal Decision Markets Suggestions

Strictly Proper Decision MarketExistence 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)