Yiling Chen Harvard Ian Kash Harvard Internet and Network Economics 2011 amitsomechgmailcom Prediction Markets Project Manager Markets used for prediction the outcome of an event ID: 238472
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Slide1
Decision Markets With Good Incentives
Yiling Chen (Harvard), Ian Kash (Harvard), Internet and Network Economics,2011.
amitsomech@gmail.comSlide2
Prediction Markets
Project Manager
Markets used for
prediction
the outcome of an event
?Slide3
Decision Markets
Using (prediction) markets for decision making.For example: Deciding between hiring Alice or Bob.
Project Manager
?Slide4
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
?
0.66
0.44Slide5
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
?
0.66
0.44Slide6
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 DDSlide7
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
?
0.66
0.44
Sequential Market yields final prediction
Decision Maker chooses an action
An outcome occurs
Scoring the expertsSlide8
Outline
What are Decision MarketsexplanationModel: notations and definitionsProblem with myopic incentives
Incentive in Decision Markets
Decision Scoring rules
Existence of a strictly proper decision market
Necessity of full support in decision scoring rulesOptimal Decision Markets SuggestionsSlide9
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 functionSlide10
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:
Slide11
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.
Slide12
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 AShorthands: d – the distribution over all actionsda
– the likelihood action a is drawn from the set AExamples:MAX:
Note that
D(
) is a distribution. We will show that it is necessary for creating myopic incentive
compatibility.
Slide13
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?
Slide14
Outline
What are Decision MarketsexplanationModel: notations and definitionsProblem with myopic incentives
Incentive in Decision Markets
Decision Scoring rules
Existence of a strictly proper decision market
Necessity of full support in decision scoring rulesOptimal Decision Markets SuggestionsSlide15
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
rule
for
the selected actionSlide16
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
?
0.66
0.44
Sequential Market yields final prediction
Decision Maker chooses an action
An outcome occurs
Scoring the expertsSlide17
So, What Is the Problem? (2)
Current Market values at some round t:Alice: 0.2 Bob: 0.8An expert with 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 0.75Slide18
So, What Is the Problem? (2)
Current Market values:Alice: 0.2 Bob: 0.8An expert with 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.Slide19
So, What Is the Problem? (3)
A. Truthful reporting:The expert raises Alice’s market value to 0.75Decision 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: PositiveRaising from 0.75 to 0.81: NegativeOverall: PositiveSlide20
So, What Is the Problem? (4)
C. Leveling Alice and Artificially Lowering Bob:The expert raises Alice’s market value to 0.75The 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: PositiveSlide21
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- 0.8Expert #2:Buying short on Alice will result in no payoffThus, Expert #2 do nothing!! Slide22
Outline
What are Decision MarketsexplanationModel: notations and definitionsProblem with myopic incentives
Incentive in Decision Markets
Decision 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 SuggestionsSlide23
Scoring Experts:
Decision Scoring RuleInstead 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.Slide24
Decision Rule:
d(P)
Decision Scoring rule:
s
o
- is a logarithmic scoring rule :1+logx
So if Alice is hired, and final prediction is
Alice:0.25, Bob:0.75
d
Alice
= 0.2,
d
Bob
=0.8
S
A
lice,finished
on time,
=5*(1+log(0.25))
S
Bob,finished
on time
,
=1.25*(1+log(0.75))
Decision Scoring Rule: ExampleSlide25
Expected score:
Q – the expert’s personal beliefP – 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 PStrictly properness: the inequality is strict unless P=Q
Decision Scoring Rule
: Slide26
Myopic Incentives in Prediction Vs. Decision Markets
Decision
Markets
Prediction
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 actionSlide27
Outline
What are Decision MarketsexplanationModel: notations and definitionsProblem with myopic incentives
Incentive in Decision Markets
Decision Scoring rules
Existence of a strictly proper decision market
Necessity of full support in decision scoring rulesWith Strictly properness, preferred action can be chosen W.P close to (but not) 1.
Optimal Decision Markets SuggestionsSlide28
Strictly Proper Decision Market
Existence of a strictly proper decision marketTheorem 1: let D be a decision rule (with full support *). Then there exists a decision rule S such that (D,S) is strictly properSlide29
Strictly Proper Decision Market (2)
Existence of a strictly proper decision marketProof:for any strictly proper scoring rule s: Then the expected payment is:
Prediction Market Scoring rule
Linearity of ExpectationSlide30
Strictly Proper Decision Market (3)
Necessity of full-supportFull support decision rule: if Slide31
This Model is Still Not Optimal
We proved that MAX decision rule can not be used in myopic incentive compatible decision marketA stochastic decision rule with full support is crucial for obtaining myopic incentive compatibilityIn practice, no decision maker will knowingly choose the wrong decision, even with small probability Slide32
Optimal Decision Markets
Right Action Rules (Chen[2012])Compensation function: (Boutilier [2012])Fool the agents (TA example)