Game Theory. Jiang, Bo ( 江波 )

Transcription

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

2 Majority voting Mechanism Design 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 Mechanism Design in Auction Auction Participants: Auctioneer Bidders Auction Types: English Auction Dutch Auction Vicrey Auction Sealed Auction

7 A Simple Auction One item; n bidders. Each player i has a scalar value that is the value he is willing to pay for this item. The winner has to pay price p The set of alternatives A (the set of possible winners)

8 No payment. How to Pay Pay your bid. Vickrey's second price auction: the winner pay the second highest price of the declared bid. Proposition(Vickrey): For every and every, let be i's utility if he bids and is his utility is he bids, then

9 Notation for General Mechanisms Formally, a mechanism needs to socially choose some alternative from A. The preference of each player i is modeled by a valuation function Social choice function Payment function

10 Incentive Compatible Mechanism A mechanism is called incentive compatible if for every player I, every and every, if we denote and then

11 Vickrey-Clarke-Groves Mechanisms A mechanism is called a Vickrey-Clarke-Groves (VCG) Mechanisms if: for some functions we have that for all Theorem(VCG): Every VCG mechanism is incentive compatible.

12 Clark Pivot Rule Individual rational: No positive transfer Clarke pivot rule: The choice is called Clarke pivot payment. A VCG mechanism with Charke pivot rule is individual rational and has no positive transfer

13 Some Examples Auction of a Single item Reverse Auction Bilateral Trade Multiunit Auctions Public Project

14 Affine Maximizer: Weighted VCG The payment function is defined as: where Mechanism is incentive compatible. Theorem(Roberts): when, is incentive compatible if and only if f is an affine maximizer.

15 Bayesian-Nash Equilibrium A simple example: Two bidder: Bob and Alice; One single item. a and b are the private values of Bob and Alice. The distribution of a and b are denoted by and respectively. x and y are the bid price respectively. Bayesian-Nash Equilibrium: In a first price auction, a, b uniformly distributed over [0,1], then the strategy x(a)=a/2 and y(b)=b/2 are in Bayesian-Nash equilibrium.

16 Revenue Equivalence Revenue get by the auctioneer is max{a/2,b/2} The revenue generated by the second price auction is min{a, b} Taking expectation on [0, 1], both yields 1/3. Bayesian-Nash implementations of the same social choice function generate the same expected revenue!

17 Any Questions?