Implementation in Bayes-Nash equilibrium

1. Implementation in Bayes-Nash equilibrium Tuomas Sandholm Computer Science Department Carnegie Mellon University

2. Implementation in Bayes-Nash equilibrium • Goal is to design the rules of the game (aka mechanism) so that in Bayes-Nash equilibrium (s1, …, s|A|), the outcome of the game is f(u1, …, u|A|) • Weaker requirement than dominant strategy implementation • An agent’s best response strategy may depend on others’ strategies • Agents may benefit from counterspeculating each others’ • preferences • rationality • endowments • capabilities … • Can accomplish more than under dominant strategy implementation • E.g., budget balance & Pareto efficiency (social welfare maximization) under quasilinear preferences …

3. dAGVA expected externality mechanism [d’Aspremont & Gerard-Varet 79; Arrow 79] • Like Groves mechanism, but sidepayment is computed based on agent’s revelation vi , averaging over possible true types of the others v-i • Outcome (x1, x2, ..., xk, m1, m2, ..., m|A| ) • Quasilinear preferences: ui(x, m) = mi + vi(x1, x2, ..., xk) • Utilitarian setting: Social welfare maximizing choice • Outcome s(v1, v2, ..., v|A|) = maxxivi(x1, x2, ..., xk) • Others’ expected welfare when agent i announces vi is (vi) = v-i p(v-i) jivj(s(vi , v-i)) • Measures change in expected externality as agent i changes her revelation • Thrm. Assume quasilinear preferences and independently drawn valuation functions vi. A utilitarian social choice function f: v -> (s(v), m(v)) can be implemented in Bayes-Nash equilibrium if mi(vi)= (vi) + hi(v-i) for arbitrary function h • Unlike in dominant strategy implementation, budget balance achievable • Intuitively, have each agent contribute an equal share of others’ payments • Formally, set hi(v-i) = - [1 / (|A|-1)] ji(vj) • Does not satisfy participation constraints (aka individual rationality constraints) in general • Agent might get higher expected utility by not participating

4. Myerson-Satterthwaite impossibility • Avrim is selling a car to Tuomas, both are risk neutral, quasilinear • Each party knows his own valuation, but not the other’s valuation • The probability distributions are common knowledge • Want a mechanism that is • Ex post budget balanced • Ex post Pareto efficient: Car changes hands iff vbuyer > vseller • (Interim) individually rational: Both Avrim and Tuomas get higher expected utility by participating than not • Thrm. Such a mechanism does not exist (even if randomized mechanisms are allowed) • This impossibility is at the heart of more general exchange settings (NYSE, NASDAQ, combinatorial exchanges, …) !

5. Proof • Seller’s valuation is sL w.p. a and sH w.p. (1-a) • Buyer’s valuation is bL w.p. b and bH w.p. (1-b). Say bH > sH > bL > sL • By revelation principle, can focus on truthful direct revelation mechanisms • p(b,s) = probability that car changes hands given revelations b and s • Ex post efficiency requires: p(b,s) = 0 if (b = bL and s = sH), otherwise p(b,s) = 1 • Thus, E[p|b=bH] = 1 and E[p|b = bL] = a • E[p|s = sH] = 1-b and E[p|s = sL] = 1 • m(b,s) = expected price buyer pays to seller given revelations b and s • Since parties are risk neutral, equivalently m(b,s) = actual price buyer pays to seller • Buyer pays what seller gets paid  ex post budget balance • E[m|b] = (1-a) m(b, sH) + a m(b, sL) • E[m|s] = (1-b) m(bH, s) + b m(bL, s) • Individual rationality (IR) requires • b E[p|b] – E[m|b]  0 for b = bL, bH • E[m|s] – s E[p|s]  0 for s = sL, sH • Bayes-Nash incentive compatibility (IC) requires • b E[p|b] – E[m|b]  b E[p|b’] – E[m|b’] for all b, b’ • E[m|s] – s E[p|s]  E[m|s’] – s E[p|s’] for all s, s’ • Suppose a=b= ½, sL=0, sH=y, bL=x, bH=x+y, where 0 < 3x < y. Now, • IR(bL): ½ x – [ ½ m(bL,sH) + ½ m(bL,sL)]  0 • IR(sH): [½ m(bH,sH) + ½ m(bL,sH)] - ½ y  0 • Summing gives m(bH,sH) - m(bL,sL)  y-x • Also, IC(sL): [½ m(bH,sL) + ½ m(bL,sL)]  [½ m(bH,sH) + ½ m(bL,sH)] • I.e., m(bH,sL) - m(bL,sH)  m(bH,sH) - m(bL,sL) • IC(bH): (x+y) - [½ m(bH,sH) + ½ m(bH,sL)]  ½ (x+y) - [½ m(bL,sH) + ½ m(bL,sL)] • I.e., x+y  m(bH,sH) - m(bL,sL) + m(bH,sL) - m(bL,sH) • So, x+y  2 [m(bH,sH) - m(bL,sL)] 2(y-x). So, 3x  y, contradiction. ■

6. Myerson-Satterthwaite impossibility… • Actually, the impossibility applies to any priors, as long as • the priors’ supports overlap, and • the priors don’t have gaps • The inefficiency is caused by two-sided private information

7. Advanced topics

8. Full implementation: virtual implementation • Here we don’t necessarily assume that payments are possible • Want to implement so that the right social choice function is achieved in all equilibria • Virtual implementation: relax this by allowing a social choice within ε to be implemented (for all ε > 0) • Thm [Serrano & Vohra GEB 2005]. Consider pure strategies only, finite type spaces, private types, and assume no-total-indifference (i.e., for each agent and each type, there are no ties in expected utility (as long as beliefs are updated using Bayes rule)). A social choice function in such environments is virtually Bayesian implementable iff it satisfies Bayesian incentive compatibility and a condition called virtual monotonicity. • Virtual monotonicity is weak in the sense that it is generically satisfied in environments with at least three alternatives • So, in most environments virtual Bayesian implementation is as successful as it can be (i.e., incentive compatibility is the only condition needed)

9. Full implementation: subgame perfection(Hart-Moore example of Moore-Repullo mechanism) Mechanism (assumes common knowledge between B and S)