Game Theory. Jiang, Bo ( 江波 )

Transcription

1 Game Theory Jiang, Bo ( 江波 ) Jiang.bo@mail.shufe.edu.cn

2 Mechanism Design in Voting Majority voting Three candidates: x, y, z. Three voters: a, b, c. Voter a: x>y>z; voter b: y>z>x; voter c: z>x>y What is the final outcome of this vote?

3 Another Example Bora rule: rank candidates according to their average ranking among voters. Two candidates : {x, y}, three voters {a, b, c}. a: x>y; b: x>y; c: y>x. Aggregated ranking: x>y. Four candidates: {x, y, z, w}, three voters {a, b, c}. a: x>y>z>w; b: x>y>z>w. How should c report his ranking to get some benefit.

4 One More Example 4 candidates : {A, B,C,D}, 30 voters 3: A > B > C > D 6: A > D > C > B 3: B > C > D > A 5: B > D > C > A 2: C > B > D > A 5: C > D > B > A 2: D > B > C > A 4: D > C > B > A

5 One More Example Majority Voting: 9 vote for A, 8 vote for B, 7 vote for C and 6 vote for D. Final outcome: A > B > C > D In the absence of C 9 vote for A, 10 vote for B, 11 vote for D. Final outcome: D > B > A

6 Basic Settings represents the order (linear order) of I L: set of linear orders A: set of candidates E: set of voters Two voting methods:

7 What is a good mechanism for social welfare fucntion Unanimity: and implies: Independence of irrelevant alternatives (IIA): implies:

8 Arrow Impossibility Theorem F is dictatorship such that for all Theorem (Arrow): When, every F satisfies unanimity and IIA is a dictatorship.

9 Properties of Social Choice Function A social choice function f can be strategically manipulated by voter i if for some with and we have that f is called incentive compatible if it cannot be manipulated.

10 Gibbard-Satterthwaite Theorem Monotone: implies that A social choice function is incentive compatible if and only if it is monotone. F is dictatorship such that for all implies that

11 Gibbard-Satterthwaite Theorem Theorem (Gibbard-Satterthwaite): Social choice function f is incentive, onto A and A 3, then f is dictatorship. Idea of the proof: construct a social welfare function F based on f, and then apply Arrow Impossibility Theorem.

12 Break Arrow Curse: Domain Restriction Single peaked preference: if there exists a point (the peak of i ) such that for all and all If the preference of each individual is single peaked and the number of voters is odd, then there is no dictator for majority voting.

13 Results on Social Choice If the preference of each individual is single peaked and the number of voters is odd. The social choice of median of peaks is incentive compatible. The social choice of kth highest peak is incentive compatible.

14 Stable Matching Dating: Can be described as matching problem Question of interest: is this matching stable?

15 Basic settings A set M of men, a set W of women. Preference ordering of agent i is, means i ranks x above y. Matching: assignment of men to women such that man is assigned at most one woman and vice versa.

16 Unstable Matching Without loss of generality, assume M = W. Matching is unstable: exists two pairs and such that: (i) is matched with (ii) is matched with (iii) and is called blocking pair.

17 An Example Is matching,, stable.?

18 Deferred Acceptance Algorithm (male-propose) Each man proposes to his top-ranked choice Each woman who has received at least two proposals keeps top-ranked proposal Each man who has been rejected proposes to his top-ranked choice among the women who have not rejected him. Again each woman who has at least two proposals (including ones from previous rounds) keeps her top-ranked proposal. The process repeats until no man has a woman to propose to or each woman has at most one proposal.

19 Property of the algorithm The algorithm converges in finite steps. The male propose algorithm terminates in a stable matching. is a matching, the woman assigned to man m is denoted by. is male-optimal if there is no stable matching such that for all m and there is at least one j satisfies.

20 Property of the algorithm The stable matching produced by the (maleproposal) Deferred Acceptance Algorithm is male-optimal. The direct mechanism associated with the male propose algorithm is strategy-proof for the males. The mechanism associated with the male propose algorithm is not strategy-proof for the females.

21 Linear Programming Formulation

22 Without Bipartite Nature Stable Roommates Problem: There is no stable roommates in the above example.

23 Any Questions?