which leave type B indifferent between deviating and playing the equilibrium strategy p(m)={∈MB2(O,m):c)=∑m1(m,r,0)p(m)} R Criterion DI says that type 0 can be eliminated for m iff there exists another type 8 such that De(m)U Do(m)C De(m): whenever 0(weakly)prefers to deviate to m 8 strictly prefers to do so Criterion D2 says that type 0 can be eliminated for m iff De(m)U Do(m)C Uo 40 De(m): whenever 0(weakly) prefers to deviate to m, there is some type 0r that strictly prefers to do so. Clearly, D2 is stronger than DI(unless there are only two types, in which case the criteria coincide. One can think of two tests derived from the criteria: first, one can require that the candidate equilibrium be supported by out-of-equilibrium beliefs which assign zero probability to eliminated types. This is rather strong, and corresponds to the SO-called "Di or D2 refinement An alternative test requires that, whenever a pair 0, 0 satisfy the condition in the definition of DI, the posterior likelihood ratio ml(m)(e) should not shift towards e that is, we require m)(⊙)p1() 1(m)()-p1(9) This leads to Divinity and related concepts. Note that, however, this class of tests is rather strong-the intuitive story is somehow. less intuitive than the one underlying the Intuitive Criterion(no pun intended ) In general, divinity and friends capture notions of "monotonicity. Finally, both Di and D2 imply equilibrium dominationwhich leave type θ indifferent between deviating and playing the equilibrium strategy: D 0 θ (m) = {ϕ ∈ MBR2(Θ, m) : u ∗ 1 (θ) = X r∈R u1(m, r, θ)ϕ(r)} Criterion D1 says that type θ can be eliminated for m iff there exists another type θ 0 such that Dθ(m) ∪ D0 θ (m) ⊂ Dθ 0(m): whenever θ (weakly) prefers to deviate to m, θ 0 strictly prefers to do so. Criterion D2 says that type θ can be eliminated for m iff Dθ(m) ∪ D0 θ (m) ⊂ S θ 06=θ Dθ 0(m): whenever θ (weakly) prefers to deviate to m, there is some type θ 0 that strictly prefers to do so. Clearly, D2 is stronger than D1 (unless there are only two types, in which case the criteria coincide.) One can think of two tests derived from the criteria: first, one can require that the candidate equilibrium be supported by out-of-equilibrium beliefs which assign zero probability to eliminated types. This is rather strong, and corresponds to the so-called “D1 or D2 refinement.” An alternative test requires that, whenever a pair θ, θ0 satisfy the condition in the definition of D1, the posterior likelihood ratio µ1(m)(θ) µ1(m)(θ 0) should not shift towards θ: that is, we require µ1(m)(θ) µ1(m)(θ 0 ) ≤ µ1(φ)(θ) µ1(φ)(θ 0 ) This leads to Divinity and related concepts. Note that, however, this class of tests is rather strong—the intuitive story is somehow... less intuitive than the one underlying the Intuitive Criterion (no pun intended.) In general, divinity and friends capture notions of “monotonicity.” Finally, both D1 and D2 imply equilibrium domination. 6