PHPE 400 Individual and Group Decision Making Eric Pacuit - - PowerPoint PPT Presentation

PHPE 400 Individual and Group Decision Making

Eric Pacuit University of Maryland

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◮ Group decision problems often exhibit a combinatorial structure. For example, selecting a committee from a set of candidates or voting on a number of yes/no issues in a referendum.

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◮ Group decision problems often exhibit a combinatorial structure. For example, selecting a committee from a set of candidates or voting on a number of yes/no issues in a referendum. ◮ In many group decision making problems, one of the alternatives is the correct one. Which group decision making method is best for finding the “correct” alternative?

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◮ Group decision problems often exhibit a combinatorial structure. For example, selecting a committee from a set of candidates or voting on a number of yes/no issues in a referendum. ◮ In many group decision making problems, one of the alternatives is the correct one. Which group decision making method is best for finding the “correct” alternative? ◮ The different issues under consideration may be interconnected.

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◮ Group decision problems often exhibit a combinatorial structure. For example, selecting a committee from a set of candidates or voting on a number of yes/no issues in a referendum. ◮ In many group decision making problems, one of the alternatives is the correct one. Which group decision making method is best for finding the “correct” alternative? ◮ The different issues under consideration may be interconnected.

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Topics

◮ Voting in Combinatorial Domains: Anscombe’s Paradox, Multiple Elections Paradox ◮ Epistemic Voting: The Condorcet Jury Theorem ◮ Judgement Aggregation

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Group of voters Assume that there are an odd number of voters Candidates: Two candidates A and B Preferences: Rank A above B Rank B above A Indifferent between A and B Aggregation method Majority rule: A wins if more voters rank A above B than B above A; B wins if more voters rank B above A than A above B;

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Group of experts Assume that there are an odd number of experts Agenda: A single proposition P Judgements: Accept P/Judge that P is true Reject P/Judge that P is false Suspend judgement about P Aggregation method Majority rule: Accept P if more people accept P than reject P; Reject P if more people reject P than accept P

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Proceduralist Justifications

“[I]dentifies a set of ideals with which any collective decision-making procedure ought to comply. [A] process of collective decision making would be more or less justifiable depending on the extent to which it satisfies them...

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Proceduralist Justifications

“[I]dentifies a set of ideals with which any collective decision-making procedure ought to comply. [A] process of collective decision making would be more or less justifiable depending on the extent to which it satisfies them...What justifies a [collective] decision-making procedure is strictly a necessary property of the procedure—one entailed by the definition of the procedure alone.”

• J. Coleman and J. Ferejohn. Democracy and social choice. Ethics, 97(1): 6-25, 1986..

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Epistemic Justifications

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Epistemic Justifications

“An epistemic interpretation of voting has three main elements: (1) an independent standard of correct decisions that is, an account of justice or of the common good that is independent of current consensus and the outcome

• f votes; (2) a cognitive account of voting that is, the view that voting

expresses beliefs about what the correct policies are according to the independent standard, not personal preferences for policies; and (3) an account of decision making as a process of the adjustment of beliefs, adjustments that are undertaken in part in light of the evidence about the correct answer that is provided by the beliefs of others. (p. 34) ”

• J. Cohen. An epistemic conception of democracy. Ethics, 97(1): 26-38, 1986.

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“Condorcet begins with the premise that the object of government is to make decisions that are in the best interest of society. This leads naturally to the question: what voting rules are most likely to yield good outcomes?....

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“Condorcet begins with the premise that the object of government is to make decisions that are in the best interest of society. This leads naturally to the question: what voting rules are most likely to yield good outcomes?.... Why should we buy the idea, though, that there really is such a thing as an

• bjectively “best” choice? Aren’t values relative, and isn’t the point of voting

to strike a balance between conflicting opinions, not to determine a correct

• ne?”
• H. P. Young. Optimal Voting Rules. The Journal of Economic Perspectives, 9:1, pgs. 51 - 64,

1995.

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◮ Group decision problems often exhibit a combinatorial structure. For example, selecting a committee from a set of candidates or voting on a number of yes/no issues in a referendum.

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◮ Group decision problems often exhibit a combinatorial structure. For example, selecting a committee from a set of candidates or voting on a number of yes/no issues in a referendum. ◮ In many group decision making problems, one of the alternatives is the correct one. Which group decision making method is best for finding the “correct” alternative?

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◮ Group decision problems often exhibit a combinatorial structure. For example, selecting a committee from a set of candidates or voting on a number of yes/no issues in a referendum. ◮ In many group decision making problems, one of the alternatives is the correct one. Which group decision making method is best for finding the “correct” alternative? ◮ The different issues under consideration may be interconnected.

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Topics

◮ Voting in Combinatorial Domains: Anscombe’s Paradox, Multiple Elections Paradox ◮ Epistemic Voting: The Condorcet Jury Theorem ◮ Judgement Aggregation

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Suppose there are equally skilled individuals, each with a probability p > 1/2

• f “choosing correctly”.

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Suppose there are equally skilled individuals, each with a probability p > 1/2

• f “choosing correctly”.

Let F be a decision method. π(F, p) is the probability of getting the answer correct, given the skills of each individual p.

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Expert rule

π(Fe, p) = p

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Majority Rule

π(Fm, p)= p3+3p2(1 − p) The probability everyone is correct is p3 The probability that 1 and 2 are correct: p2(1 − p) The probability that 2 and 3 are correct: p2(1 − p) The probability that 1 and 3 are correct: p2(1 − p)

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Majority Rule

π(Fm, p)= p3+3p2(1 − p) The probability everyone is correct is p3 The probability that 1 and 2 are correct: p2(1 − p) The probability that 2 and 3 are correct: p2(1 − p) The probability that 1 and 3 are correct: p2(1 − p)

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Majority Rule

π(Fm, p)= p3+3p2(1 − p) The probability everyone is correct is p3 The probability that 1 and 2 are correct: p2(1 − p) The probability that 2 and 3 are correct: p2(1 − p) The probability that 1 and 3 are correct: p2(1 − p)

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Majority Rule

π(Fm, p)= p3+3p2(1 − p) The probability everyone is correct is p3 The probability that 1 and 2 are correct: p2(1 − p) The probability that 2 and 3 are correct: p2(1 − p) The probability that 1 and 3 are correct: p2(1 − p)

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• Theorem. When there are three voters, each with a probability p > 1/2 of

choosing correctly, then majority rule is preferred to the expert rule.

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• Theorem. When there are three voters, each with a probability p > 1/2 of

choosing correctly, then majority rule is preferred to the expert rule.

• Theorem. Assume p1 ≥ p2 > p3 > 1/2, then the simple majority rule is

preferred to the expert rule.

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Consider 3 votes, each with a confidence level p = 2/3.

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Consider 3 votes, each with a confidence level p = 2/3. The probability of at least m voters being correct is:

n

• h=m

n h

• ∗ ph ∗ (1 − p)n−h

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Consider 3 votes, each with a confidence level p = 2/3. The probability of at least m voters being correct is:

n

• h=m

n h

• ∗ ph ∗ (1 − p)n−h

3 2

• ∗ (2/3)2 ∗ 1/31 +

3 3

• 2/33 ∗ 1/30

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Consider 3 votes, each with a confidence level p = 2/3. The probability of at least m voters being correct is:

n

• h=m

n h

• ∗ ph ∗ (1 − p)n−h

3 2

• ∗ (2/3)2 ∗ 1/31 +

3 3

• 2/33 ∗ 1/30

= 3 ∗ 4/27 + 1 ∗ 8/27 = 20/27

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Condorcet Jury Theorem

State of the world x takes values 0 and 1 Ri is the event that voter i votes correctly. Mn is the event that a majority of n member electorate votes correctly.

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Condorcet Jury Theorem

State of the world x takes values 0 and 1 Ri is the event that voter i votes correctly. Mn is the event that a majority of n member electorate votes correctly. Independence R1, R2, . . . are independent conditional on x Competence: for each x ∈ {0, 1}, Pr(Ri | x) > 1

2

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Condorcet Jury Theorem

State of the world x takes values 0 and 1 Ri is the event that voter i votes correctly. Mn is the event that a majority of n member electorate votes correctly. Independence R1, R2, . . . are independent conditional on x Competence: for each x ∈ {0, 1}, Pr(Ri | x) > 1

2

Condorcet Jury Theorem. Suppose Independence and Competence. As the group size increases, the probability Pr(Mn) that a majority votes correctly (i) increases and (ii) converges to one.

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P(M) =

n

• k=(n+1)/2

n k

• pk(1 − p)n−k

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∆ = P(M) − p

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• D. Austen-Smith and J. Banks. Aggregation, Rationality and the Condorcet Jury Theorem. The

American Political Science Review, 90, 1, pgs. 34 - 45, 1996.

• D. Estlund. Opinion Leaders, Independence and Condorcet’s Jury Theorem. Theory and Decision,

36, pgs. 131 - 162, 1994.

• F. Dietrich. The premises of Condorcet’s Jury Theorem are not simultaneously justified. Episteme,

2008.

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Correlated Votes

Votes can also be correlated in many ways: ◮ Any common cause of votes is a potential source of dependence. For example, common evidence, such as, in a court case, witness reports and the defendant’s facial expression, or, among scientists, experimental data. ◮ Voters can also be influenced by common causes that are non-evidential such as distracting heat: such causes lack an objective bearing on the true state, and yet they influence people’s epistemic performance and thereby threaten independence.

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• U. Hahn, M. von Sydoow and C. Merdes. How communication can make voters choose less well,

Topics in Cognitive Science, 11(1), 194 - 206, 2018. .

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What happens if there are more than two options?

• C. List and R. E Goodin. Epistemic democracy: Generalizing the Condorcet jury theorem. Journal
• f political philosophy, 9(3):277–306, 2001.

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7 Propositions

0.0 0.2 0.4 0.6 0.8 1.0 20 40 60 80 100 20 40 60 80 100 0.2 0.4 0.6 0.8 1.0

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