Mechanism Design for Federated Sponsored Search Auctions

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5 To prove that the ex post IC does not hold, consider the specific case in which x = x 1. In the case both agents report truthfully their type, s 2 s expected utility is E[u s2 (θ)] = c a1 q s2 v s2 (given that her payment is zero). In the case s 2 misreports her report such that the resulting efficient allocation is x 2, her expected utility is E[u 2 (ˆθ)] = q s2 v s2 (given that her payment is zero). Since q s2 v s2 c a1 q s2 v s2, s 2 strictly prefers to misreport her type and therefore there cannot be an incentive compatible mechanism given AE, ex interim WBB, and ex interim IR. To overcome the impossibility, we propose a VCG execution contingent mechanism in which the payments depend on the actual realization of the events. More precisely, the allocation function is efficient and is defined as in Equation (1) and the payments are defined as: p s (ˆθ ω) = α a (f(ˆθ s ), ˆQ) ˆv a ˆv a a A\A s a A\A s,a ω Notice that the payments depend on the ads actually clicked, captured in the formula by ω Ω. Theorem 5. For the FSE model with externalities the above execution contingent VCG is AE, ex post IC, ex interim IR, and ex interim WBB. Proof. Assume that all the agents but s truthfully report their report values, i.e., V s = ˆV s and Q s = ˆQ s. Then, if agent s report ˆV s and ˆQ s, its expected utility is: E[u s (ˆθ, θ)] = α a (f(ˆθ), ( ˆQ s,q s )) v a a A α a (f(ˆθ s ), ( ˆQ s,q s ))ˆv a 0 a A/A s First, we note that there is nothing that agent s can do to manipulate the second term of the expected utility, given the reports of other agents. The reason is that this part of the expected utility of s is independent of her reported values. Secondly, we note that the first term of expected utility is computed after the observation of the realization of the event. While the selection of x depends on ˆV and ˆQ, the actual outcome upon the realization depends on V and Q. As a result, note that, by definition of f(ˆθ): α a (f(ˆθ), (Q s,q s )) v a a A α a (f(ˆθ), ( ˆQ s,q s )) v a a A Thus, if all agents s truthfully report their valuations and qualities, agent s is also best off revealing its information truthfully, since this will result in a mechanism selecting the allocation which maximize the social welfare in expectation. This leads to the expected utility maximization of agent s. The execution contingent mechanism is ex interim IR, since a A α a(f(ˆθ),q) v a a A α a(f(ˆθ s ),Q s ) v a, implying that E[u s (ˆθ, θ)] 0. It can be easily seen that it is also ex interim WBB. We now show that the proposed mechanism cannot guarantee ex post IR and ex post WBB. Example. Consider a situation with two agents (search engines) s 1 and s 2, and two available slots (i.e., x =2). Let A = {a 1,a 2,a 3 }, and A s1 = {a 1,a 2 } and A s2 = {a 3 }.We compute the agents payments for each possible realization according to the above mechanism: p s1 (ˆθ ) =α a3 (f(ˆθ s2 ), ˆQ)ˆv a3 p s1 (ˆθ {a 3}) =α a3 (f(ˆθ s2 ), ˆQ)ˆv a3 ˆv a3 p s2 (ˆθ ) =α a1 (f(ˆθ s1 ), ˆQ)ˆv a1 + α a2 (f(ˆθ s1 ), ˆQ)ˆv a2 p s2 (ˆθ {a 1}) =α a1 (f(ˆθ s1 ), ˆQ)ˆv a1 + α a2 (f(ˆθ s1 ), ˆQ)ˆv a2 ˆv a1 p s2 (ˆθ {a 2}) =α a1 (f(ˆθ s1 ), ˆQ)ˆv a1 + α a2 (f(ˆθ s1 ), ˆQ)ˆv a2 ˆv a2 p s2 (ˆθ {a 1,a 2}) =α a1 (f(ˆθ s1 ), ˆQ)ˆv a1 + α a2 (f(ˆθ s1 ), ˆQ)ˆv a2 2 ˆv ai Consider the joint realization in which no displayed ads is clicked, i.e., ω =. The utility of s 1 is: u s1 (ˆθ, θ ω) = α a3 (f(ˆθ s2 ), ˆQ)ˆv a3 It is actually negative and therefore ex post IR is not satisfied. Consider the joint realization in which all the displayed ads are clicked, i.e., ω = {a 1,a 2,a 3 }. The utility of FSE is: E[u F (ˆθ)] = ˆv a (1 α a (f(ˆθ s ), ˆQ)) s S a S s Both agents payments are strictly negative (except when CTRs are zero) and therefore ex post WBB is not satisfied. The above example shows that, in addition to ex post IR (as shown by Theorem 2), also ex post WBB may be not satisfied. This may not be desirable because the FSE could, in specific instances, run a deficit. However, we show that this limitation is not specific to our mechanism but due to the nature of the problem. To prove this, we show in the next theorem that no mechanisms with WBB in ex-post can be implemented. Theorem 6. For the FSE model with externalities there is no execution contingent mechanism that is AE, ex post IC, ex post WBB, and ex interim IR. Proof. It can be easily observed that only efficient execution contingent mechanisms that compute ex post payments (w.r.t. realizations) such that the payments in expectation are those prescribed by Groves mechanisms can be incentive compatible. We recall that Groves payments are defined as E[p s (ˆθ)] = h s (ˆθ s ) a A\A s α a (f(ˆθ s ), ˆQ) ˆv a where h s is an arbitrary function, while execution contingent payments are defined as E[p s (ˆθ)] = ω Ω p s(ˆθ ω) ρ(ω) where ρ(ω) is the probability associated to the realization ω. Thus, in order to prove the theorem, we just need to show that there is not any efficient execution contingent mechanism whose payments are WBB in ex post and are equivalent in expectation to those of some Groves mechanism. We show it by a counterexample. Consider a situation with two search engines S = {s 1,s 2 } and two available slots. Let A 1 = {a 1 }, A 2 = {a 2 }. Consider the specific case of the cascade model. The report are: q a1, c a1, v a1, q a2, c a2, v a2 where c ai is a function of allocation x and qualities Q. To prove that ex post WBB does not hold, consider the case q a1 v a1 + c a1 q a2 v a2 q a2 v a2 + c a2 q a1 v a1. i=1 612

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