ht
P(rt=buyrec|xt=0, ht, B)≤1−1−qG
qB (30)
and sup
ht P(rt=sellrec|xt=1, ht, B)≤1−1−qG
qB . (31)
Proof. Our objective will be to construct a mimicking strategy recursively, starting with t=1.
Let
P(r1=buyrec|h1, x1=1, B)=1=P(r1=sellrec|h1, x1=0, B). (32) As v1=12, then (32) implies (27) (with h1substituting ht). That (32) implies (28) and (29) is immediate.
Next suppose a strategy can be constructed such that, up to period T−1 ≥1, for all ht with t≤T −1, equation (27) and at least one of (28) and (29) are satisfied. Now fix a history hT; we will consider 3 cases separately.
Case 1:if
vBTqB+(1−vBT)(1−qB)=vTGqG+(1−vTG)(1−qG), let
P(rT =buyrec|hT, xT =1, B)=1=P(rT =sellrec|hT, xT =0, B).
Case 2:if instead
vBTqB+(1−vBT)(1−qB) < vTGqG+(1−vTG)(1−qG), (33) let
P(rT =buyrec|hT, xT =1, B)=1,
and
P(rT =buyrec|hT, xT =0, B)=z(hT), with z(hT)solving
vTB[qB+(1−qB)z] +(1−vBT)[(1−qB)+qBz] =vTGqG+(1−vTG)(1−qG). (34) The right-hand side of (34) is bounded above by qG, and qG<1; the left-hand side evaluated at z=1 is equal to 1; lastly, the left-hand side evaluated at z=0 is strictly smaller than the right-hand side (by (33)). So z(hT)exists (and is unique, moreover).
Case 3:if
vTBqB+(1−vBT)(1−qB) > vTGqG+(1−vTG)(1−qG), (35) let
P(rT =sellrec|hT, xT =0, B)=1, and
P(rT =sellrec|hT, xT =1, B)=z(hT), with z(hT)solving
vTB[qBz+(1−qB)] +(1−vBT)[(1−qB)z+qB] =vTG(1−qG)+(1−vTG)qG. (36) The right-hand side of (36) is bounded above by qG, and qG<1; the left-hand side evaluated at z=1 is equal to 1; lastly, the left-hand side evaluated at z=0 is strictly smaller than the right-hand side (by (35)). So z(hT)exists (and is unique, moreover).
It is now a simple matter to check that, irrespective of which of the cases above holds, then given hT, equation (27) and at least one of (28) and (29) are satisfied. As hT was chosen arbi-trarily, a strategy can therefore be constructed such that for all ht with t≤T, equation (27) and at least one of (28) and (29) are satisfied. But then a mimicking strategy exists, by induction.
We show next that the mimicking strategy constructed above satisfies (30). In Cases 1 and 3, P(rt=buyrec|xt=0, ht, B) =0, so (30) trivially holds. In Case 2, P(rt =buyrec|xt = 0, ht, B) =z(ht), with z(ht)solving (34). So we need to show z(ht) ≤1 −1−qBqG. Notice that the left-hand side of (34) is increasing in zand, as qB≥12, is also increasing in vtB. The right-hand side of (34) is bounded above by qG. Since vBt ≥0, the solution to
(1−qB)+qBz=qG
is thus an upper bound for z(ht). This gives z(ht) ≤1 −1−qBqG. The mimicking strategy con-structed above therefore satisfies (30). That it satisfies (31) follows by symmetry. 2
Define in the remaining of the appendix, for any non-degenerate mixed strategy of the B analyst,
Lθ,t:=P(ht|V =0, θ ) P(ht|V =1, θ ) and
φθ,tr :=P(rt=r|V =0, ht, θ ) P(rt=r|V =1, ht, θ ).
Lemma 9. For any non-degenerate mixed strategy of the Banalyst and any history ht such that vGt (sellrec) =vBt (sellrec)and vGt (buyrec) =vtB(buyrec):
vBt (buyrec)−vtB(sellrec)
vGt (sellrec)−vtB(sellrec)= (1+φsellG,tLG,t)(φB,tsell−φB,tbuy) (1+φB,tbuyLB,t)(φB,tsell−φG,tsellLLG,t
B,t); (37)
vtB(buyrec)−vBt (sellrec)
vGt (buyrec)−vBt (buyrec)= (1+φbuyG,tLG,t)(φB,tsell−φB,tbuy) (1+φB,tsellLB,t)(φbuyB,t−φG,tbuyLLG,t
B,t); (38)
vtG(buyrec)−vGt (sellrec)
vtG(buyrec)−vBt (buyrec)= (1+φB,tbuyLB,t)(φG,tsell−φbuyG,t) (1+φG,tsellLG,t)(φB,tbuyLLB,t
G,t −φbuyG,t); (39)
vGt (buyrec)−vBt (buyrec)
vtG(sellrec)−vBt (sellrec)=(1+φG,tsellLG,t)(1+φB,tsellLB,t)(φbuyB,t−φbuyG,tLLG,t
B,t) (1+φG,tbuyLG,t)(1+φB,tbuyLB,t)(φB,tsell−φsellG,tLLG,t
B,t)
. (40)
Proof. Applying Bayes’ rule, vθt(r)= 1
1+φθ,tr Lθ,t.
Tedious but straightforward algebra then yields (37)–(40). 2 Lemma 10. If Bmimics G, (30)–(31)hold, and qB>12 then
infht
φsellB,t−φB,tbuy>0. (41) Furthermore, if lim
t→∞nt= +∞, then
tlim→∞LB,t=0. (42)
Proof. Assume qB>12, Bmimics Gand (30)–(31) hold. Fix a history ht. Either (28) holds, or (29) does. We consider below the two possibilities.
If (28) holds then P(rt=buyrec|xt=0, ht, B) =z(ht), with z(ht)solving (34). Hence, P(rt=buyrec|V =1, ht, B)−P(rt=buyrec|V =0, ht, B)
=qB+(1−qB)z(ht)− [(1−qB)+qBz(ht)]
=(2qB−1)(1−z(ht))
≥(2qB−1)(1−qG)
qB ,
where we used (30) to obtain the final inequality.
If instead (29) holds then P(rT =sellrec|hT, xT =1, B) =z(hT), with z(ht)solving (36).
Hence,
P(rt=buyrec|V =1, ht, B)−P(rt=buyrec|V =0, ht, B)
=qB(1−z(ht))−(1−qB)(1−z(ht))
=(2qB−1)(1−z(ht))
≥(2qB−1)(1−qG) qB
, where we used (31) to obtain the final inequality.
As htwas chosen arbitrarily, we find infht
For the second part of the lemma, write LB,t=P(ht|V =0, B) Proof of Theorem2: Suppose Bmimics G(we are not saying at this point that such a strategy is optimal for B). Note to start with that if we were able to show that, for any t and any history ht,
mins vt(buyrec, s) > pat; (44)
maxs vt(sellrec, s) < pbt, (45)
then we would deduce (by induction) (a) P(ρt=ρ1) =1 for all tand ht, and also (b) mimicking Gis an optimal strategy of the B analyst. So to prove the theorem we just need to show that (44)–(45) can be made to hold for all t and all ht under the premises that B mimics G and ρt =ρ1 irrespective of t and ht. Now consider >0. As long as γG∈(0, 1), we can choose η >0 such that γB> γG−ηimplies
maxr,s |ρ1(r, s)−ρ1|< ε. (46)
Our goal in the rest of the proof will be to show that if (i) Bmimics G,
(ii) (30)–(31) hold, (iii) (46) holds,
(iv) ρt=ρ1for all tand ht,
then (44)–(45) hold for all t and ht as long as we choose εsufficiently small. By virtue of the previous remarks, this will prove the theorem.
Assume qB>12, γG∈(0, 1)and (i)-(iv) listed above. Below, we focus on histories for which vGt (buyrec) > vtB(buyrec)and vGt (sellrec) > vBt (sellrec)(the other cases are analogous). In this case (44)–(45) hold if we can show the following inequalities:
(ρ1−)vtG(buyrec)+(1−ρ1+)vtB(buyrec) > pat; (47) straightforward algebra now shows that we can write (47)–(48) as
vt(buyrec)−vt(sellrec) histories. Furthermore, Lemma10shows that lim
nt→+∞LB,t =0 and repeating arguments used in the proof of Theorem1establishes lim
nt→+∞LG,t= lim
nt→+∞
LG,t
LB,t =0. Hence, applying Lemmata9 and 10, positive number. As αt is bounded away from 0 and 1, choosing εsufficiently small assures that (51) holds. Repeating the steps above starting from (50) instead of (49) shows in a similar manner that (50) holds as long as we choose εsufficiently small. 2
Appendix C. Proofs of Section6
Proof of Proposition5. The method of proof is similar to the method used to prove Theorem2.
Suppose Bmimics G(we are not saying at this point that such a strategy is optimal for B). Note to start with that if we were able to show that, for any tand any history ht,
mins vt(buyrec, s) > pat; (52)
maxs vt(sellrec, s) < pbt, (53)
then we would deduce (by induction) (a) P(ρt=ρ1) =1 for all tand ht, and also (b) mimicking G is an optimal strategy of the B analyst. So to prove the proposition we just need to show that (52)–(53) can be made to hold for all t and all ht under the premises that Bmimics Gand ρt=ρ1irrespective of t and ht. Now consider >0. Choosing ρ <1 sufficiently large, ρ1> ρ implies
maxr,s |ρ1(r, s)−ρ1|< ε. (54)
Our goal in the rest of the proof will be to show that if (i) Bmimics G,
(ii) (54) holds,
(iii) ρt=ρ1for all t and ht,
then (52)–(53) hold for all t and ht as long as we choose εsufficiently small. By virtue of the previous remarks, this will prove the proposition.
Assume (i)-(iii) listed above. Below, we focus on histories for which vGt (sellrec) >1/2 (the other cases are analogous). In this case (52)–(53) hold if we can show the following inequalities:
(ρ1−)vGt (buyrec)+(1−ρ1+)1 straightforward algebra now shows that we can write (55)–(56) as
ρ1
Next, denote vGt (r)the valuation conditional on θ=G, rt=rand Vt=Vt−1. We can then write vGt (buyrec)−vtG(sellrec)=(1−z)[vGt (buyrec)−vGt (sellrec)]
+z[v1G(buyrec)−v1G(sellrec)].
This gives vGt (buyrec) −vGt > z[v1G(buyrec) −v1G(sellrec)] >0. The left-hand sides of (57) and (58) are therefore bounded below by a strictly positive number. As αt is bounded away from 0 and 1, choosing εsufficiently small assures that (57)–(58) hold. 2
Appendix. Supplementary material
Supplementary material related to this article can be found online at https://doi .org /10 .1016 / j .jet .2019 .01 .001.
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