chapter 23 incentives and mechanism design appendix2Word

　　Full Implementation with Symmetric Information

　　Motivation King Solomon s problem (from Moore 1990): Alice and Bess dispute maternity of a baby. Solomon suggests to cut the baby in half. One woman says: instead give it to the other. Solomon: this must be the true mother, give baby to her. Note: the mechanism is not well-defined - what if both said “give it to the other”? Is there a mechanism that ensures that the baby goes to the true mother?

　　In this case have symmetric information. Have an IC mechanism: both report the state, kill both if disagree. Truth telling is a NE. But so is lying. So we want full implementation

　　Context of implementation All agents know state of the world θ, the designer does not Or agents themselves write mechanism/contract before learning state , θ contract executed after everyone learns θ. Specifying mechanism in advance could give better outcome for ex ante risk sharing/investment incentives.

　　Formalization of King Solomon s Problem two states : α - A is the mother β - B is the mother Four outcomes: a - give baby to A b - give baby to B c - cut in half d - death to everyone

　　Preference rankings in the two states:α β

　　RAa

　　RBb

　　RAa

　　RBb

　　bc d

　　ca d

　　cb d

　　ac d

　　Can we design a mechanism whose unique equilibrium gives outcome a in state α and outcome b in state β?

　　In state α , Nash equilibrium message profile must implement a, and B can t have a deviation message that yields b or c. But then in state β, B will also not want to deviate. And A will obviously not want to deviate either gets her preferred outcome. Thus the same message profile will be Nash equilibrium in state β as well. Formally, the social choice rule is not monotonic

　　Nash Implementation State of the world R ∈ Θ describes preference relations R1,…, RI on X. choice rule (correspondence) F : Θ X Mechanism Γ= {M1,M2,…,MI ; g(.)}, g : M1 M2 … MI - X Set of Nash Equilibria:

　　Γ (fully) Nash implements F if g(EN (Γ;R)) = F (R) for any R ∈ Θ

　　Which choice rules are fully Nash implementable ? Can t use the revelation principle! Let L (x, Ri) = {y ∈ X : xRiy} EN (Γ;R)={m ∈ M1…MI: Bi (m) L(g(m),Ri) for any i in I } Here, Bi (m) ={g(m i,m-i): m i ∈ Mi} is the accessible set of agent i.

　　Monotonic (Maskin 1999) F is monotonic if for any R,R in Θ and any x in X, x ∈ F (R) , L (x;Ri) L (x;R i) for any i in I , We have x ∈ F (R ) Intuition: if x was optimal and it goes up (weakly) in all agents preference rankings, it must remain “optimal”. Some notion of independence of irrelevant alternatives.

　　Proposition 1 Any Nash implementable choice rule is monotonic

　　Examples of monotonic choice rules Pareto optimality Counterexamples: King Solomon s problem: F (α) = {a} and a goes up (weakly) in both preference rankings from α to , yet a F (

　　β).

　　no veto power condition F has no veto power if for any x in X and any R in Θ ,we have

　　Maskin theorem With I ≥ 3, any choice rule that is monotonic and has no veto power is Nash implementable. Proof by construction. Use all the information and discriminate the choice.