Online Appendix for “Sequential preference revelation in incomplete information settings” † James Schummer∗ and Rodrigo A. Velez May 6, 2019 1 Proof of Theorem 2 Theorem 2 is implied by the following result. Theorem. Let f be a strategy-proof, non-bossy SCF, and fix a reporting order Λ ∆(Π). Suppose that at least one of the following conditions holds. 2 1. The prior has Cartesian support (µ Cartesian) and Λ is deterministic. 2 M 2. The prior has symmetric Cartesian support (µ symm Cartesian) and f is weakly anonymous. 2 M − Then equilibria are preserved under deviations to truthful behavior: for any N (σ,β) SE Γ (Λ, f ), ,µ , 2 h U i (i) for each i N, τi is sequentially rational for i with respect to σ i and βi ; 2 − (ii) for each S N , there is a belief system such that β 0 ⊆ N ((σ S ,τS ),β 0) SE β 0(Λ, f ), ,µ . − 2 h U i Proof. Let f be strategy-proof and non-bossy, Λ ∆(Π), and µ Cartesian. Since the conditions of the two theorems are the same,2 we refer to2 arguments M ∗Department of Managerial Economics and Decision Sciences, Kellogg School of Manage- ment, Northwestern University, Evanston IL 60208; [email protected]. †Department of Economics, Texas A&M University, College Station, TX 77843; [email protected]. 1 made in the proof of Theorem 1. In particular all numbered equations refer- enced below appear in the paper. N Fix an equilibrium (σ,β) SE Γ (Λ, f ), ,µ . The first claim of the theorem— 2 h U i sequential rationality of τi w.r.t. (σ,β)—can be derived from inequalities estab- lished in the proof of Theorem 1. Specifically, consider the case of Condition 1 (the prior has Cartesian support and Λ is deterministic). The proof establishes the equality of (7) and (8) (see (12)). That is, the expected payoff from a truthful report is equal to the expected payoff from an equilibrium report, condition- ing on the agents’ interim belief. Since the equilibrium report is sequentially rational, so is a truthful report. In the case of Condition 2 (µ symm Cartesian, weakly anonymous f ), the analogous arguments are made via2 M (7 ), (8−), and (12 ). 0 0 0 To prove the second claim of the theorem, it suffices to prove the single- ton case S i . Repeated application of this statement proves the general ≡ f g result for arbitrary S. Let σ˜ (σ i ,τi ). First we construct a belief system γ and ≡ − N demonstrate its consistency. Lastly we show (σ˜ ,γ) SE Γ (Λ, f ), ,µ . 2 h U i & Consistency. Let (&,β ) be an assessment where & is an arbitrary profile with full support and β & is the unique belief system obtained by Bayesian updating " " given &, Λ, and µ. For any " > 0, let (&,β ) be the assessment where σ is the " " (full support) strategy profile σ (1 ")σ˜ + "& and β is the unique belief system obtained by Bayesian updating,≡ − given σ", Λ, and µ. Clearly σ" σ˜ . i ! More specifically, for each i N , each ui , each ht H , and each vi , " 2 2 U 2 2 U we have σ ui ,ht (vi ) σ˜ ui ,ht (vi ) as ε 0. We defineh γ toi be the! Bayesianh i update of!σ˜ (when well defined) or equal to " β (otherwise). That is, fix i N and any admissible ui (i.e. occurring with positive probability under µ2). 2 U i For each ht H that occurs with positive probability given σ˜ and Λ, and • 2 for each admissible (π, u i ), let γi ui ,ht (π, u i ) be defined by Bayesian updating given σ˜ and Λ. − h i − i For each ht H that occurs with zero probability given σ˜ and Λ, and for • 2 & each admissible (π, u i ), let γi ui ,ht (π, u i ) β ui ,ht (π, u i ). − h i − ≡ h i − Using Bayes’ rule, one can write an explicit expression of β " in terms of ", σ˜ , &, Λ, and µ. We omit this expression since it is easy to see that β " γ; specifically, for each i N , each u , each h H i , each , and each! v N i , i t π Π i nf g 2 2 U 2 2 − 2 U 2 " βi ui ,ht (π, v i ) γi ui ,ht (π, v i ) as " 0. Thus (σ˜ ,γ) is a consistent as- h i − ! h N i − ! sessment for Γ (Λ, f ), ,µ . h U i Sequential rationality. We use the notation—from Case 2 of the proof of The- orem 1—where f (ht , vπ t 1,...,n π ) represents the outcome of f when the or- 0( + ) 0 dered reports in h are made accordingj to the agents’ order under . We also t π0 refer to Equations (2 )–(13 ) in order to prove various claims. For the simpler 0 0 case that Λ is deterministic, the analogous equations from Case 1 apply. We show that σ˜ is sequentially rational for beliefs γ. That is, for each Agent j , σ˜ j prescribes a report that maximizes j ’s expected payoff after any history feasible for j , given σ˜ j and γj . − Case j = i (σ˜ j = τi ). For Agent i , we show the sequential rationality of truth- telling using (2 ) which states that, regardless of the history, continuation strate- 0 gies under σ are welfare-equivalent to truthful ones. Since σ˜ i = σ i the result follows. − − To formalize this, fix any t 1,...,n , π supp(Λ) with π(t ) = i , ht 1 2 f g 2 − 2 Hi , and u supp(µ), and recall that σ˜ i = τi . For any vi0 supp(σ ht 1, ui ), 2 2 h − i consider the two t -period histories (ht 1, vi0) and (ht 1, ui ). For any two σ- continuations of those two histories given− π, namely for− any1 v π(t +1,...,n) with σ v h , v ,π, u > 0, and π0 (t +1,...,n) ( π0 (t +1,...,n) ( t 1 i0) π(t +1,...,n)) • 2 U j − v π(t +1,...,n) with σ v h , u ,π, u > 0, π00(t +1,...,n) ( π00(t +1,...,n) ( t 1 i ) π(t +1,...,n)) • 2 U j − applying (2 ) to histories h and h , u respectively yields 0 t 1 ( t 1 i ) − − u f h , v , v π u f h , u π , and ( ( t 1 i0 π0 (t +1,...,n) )) = ( ( t 1 π(t ,...,n) )) − j − j u f h , u , v π u f h , u , u π . (**) ( ( t 1 i π00(t +1,...,n) )) = ( ( t 1 i π(t +1,...,n) )) − j − j Observe that the two RHS’s are equivalent and thus i receives the same payoff from reporting ui as from reporting vi0. Since reporting vi0 maximizes i ’s ex- pected payoff after ht 1 given ui and σ i , reporting ui maximizes i ’s expected − − payoff after ht 1 given ui and σ˜ i = σ i , proving sequential rationality. More generally, however,− all agents receive− the− same payoff after i ’s deviation from σi to truthful τi . This is true ex post of any realization of π (with π(t ) = i ), which is relevant in the next case. 1 If t = n these subprofiles are null lists, and the proof simplifies. 3 Case j = i (σ˜ j = σj ). The intuition behind this case is that, since Equation (**) implies6 that i ’s deviation to truth-telling does not change continuation pay- offs, the incentive compatibility conditions of the original equilibrium (σ) are preserved. Fix any t 1,...,n , π supp(Λ) with π(t ) = j , ht 1 Hi , u supp(µ), 2 f g 2 − 2 2 and vj supp(σj ht 1, u j ). We wish to show that vj maximizes j ’s expected 2 h − i payoff, given σ˜ j and γ. We begin with− the more difficult subcase that 1 i t , so i acts after j . π− ( ) > Denote the (possibly empty) sets of agents who act between j and i and after i as 1 1 1 B = k N : π− (j ) < π− (k) < π− (i ) f 2 1 1 g A = k N : π− (i ) < π− (k) f 2 g Fix a (deviation) report vj0 . We shall compare payoffs obtained under four profiles of reports, 2 U (ht 1, vj , vB , vi , vA) − (ht 1, vj , vB , ui , wA) − (ht 1, vj0 , vB0 , vi0, vA0 ) − (ht 1, vj0 , vB0 , ui , wA0 ) − where the various subprofiles for B, i , and A satisfy vB supp(σB (ht 1, vj ), uB ) 2 h − i vB0 supp(σB (ht 1, vj0 ), uB ) 2 h − i vi supp(σi (ht 1, vj , vB ), ui ) 2 h − i vi0 supp(σi (ht 1, vj0 , vB0 ), u ) 2 h − i vA supp(σA (ht 1, vj , vB , vi ), uA ) 2 h − i wA supp(σA (ht 1, vj , vB , ui ), uA ) 2 h − i vA0 supp(σA (ht 1, vj0 , vB0 , vi0), uA ) 2 h − i wA0 supp(σA (ht 1, vj0 , vB0 , ui ), uA ) 2 h − i 4 Equation (**) implies the following two equalities. u(f (ht 1, vj , vB , vi , vA π)) = u(f (ht 1, vj , vB , ui , wA π)) − j − j u(f (ht 1, vj0 , vB0 , vi0, vA0 π)) = u(f (ht 1, vj0 , vB0 , ui , wA0 π)) − j − j Sequential rationality of σ implies u j (f (ht 1, vj , vB , vi , vA π)) u j (f (ht 1, vj0 , vB0 , vi0, vA0 π)).