In the previous sections we have created a basic model for information transmission in hierarchies.
While there may be several ways to extend this model, we want to discuss two important extensions in this section. The first extension relaxes the assumption we made on the rewardR. In the second extension we introduce shared information.
77
2.4.1 Extension 1: Unbound reward
For the entire paper we were assuming that the rewardRis not larger thann·(xn−1−xn) + (n− 2)·xn−1. This assumption ensures that even in a setting with just two periods, all different types of SPNE exist, i.e. n·(xn·xn+R
n+R) < 2·x2·xn−1
n−1+R holds. If we relax this assumption and select a larger reward, this is no longer true. The equilibrium in each period and the SPNE will no longer be unique, unless we specify an equilibrium selection rule.
The only equilibrium that can coexist with another equilibrium for the same values of the param-eters is the equilibrium in which all players pass in the first period. At the same time there is no more interval in which just one player holds her information in the first period. For higher values ofδ, but which are less than n·(xn·xn+R
n+R), there will be two equilibria: On the one hand the same as before, where some players hold their information in the first period, and in addition the equilib-rium in which all players pass int = 1. If Ris very large and if the discount factor is close to 1, there can even be a region in which there is an equilibrium with all players holding in the first period and at the same time an equilibrium with all players passing in the first period.
Naturally two different ways of equilibrium selection exist. We can either always select the equi-librium where more players hold or where more players pass. In the first case the analysis stays quite similar to the one we did with the assumption onR. The only difference is that in a two-layer model the duration changes directly from a single period toT periods, but there is no interval for δwith a duration of2periods.
This is not the case if we always select the equilibrium in which all players pass. For values of δ larger than n·(xn·xn+R
n+R), some players will hold their information in the first period. In this new setting Proposition 2.2 no longer holds. This means that even if two or more players hold their information int= 1, they will not necessarily wait untilT to pass. In this case we get additional conditions for the SPNE, which describe the players behavior starting from the second period.
Furthermore, the duration not necessarily will be strictly increasing inδ.
We also need similar conditions for other equilibrium selection methods, as long as not always the equilibrium where most players hold gets selected.
2.4.2 Extension 2: Shared information
While in some examples the uniqueness of the information of the different players comes naturally, one can also argue that in some cases there should be shared information. A researcher might propose the same idea for different projects, or a mathematical proof gives a stronger result than necessary.
For simplification we will focus on a two-layer model again. We include the possibility that two players have some shared information in addition to their unique information. Only the player who passes the information first gets the payoff according to the value of the shared information.
We assume that if both players pass at the same time, they split the payoff equally. In Section 2.2.4 we have described the impact of an increase in the value of information on the duration. We have seen that only if the value of one of the two least informed players is changed, the duration is affected. In this situation the result will be similar. If at least one of the players n−1andn
78 Chapter 2. Information Transmission in Hierarchies has shared information, the duration may decrease. Still the effect is different than an increase in xn(orxn−1), because the players do not get this additional payoff for sure. We still assume that if several players have an incentive to pass the information, the players with the more valuable unique information pass first.
Let playerjand khave the shared information with value y >0. Without loss of generality we can assume thatxj ≥ xkholds. To have a duration of a single period all players need to prefer passing over holding in the first period, i.e.
∀i∈N \ {j, k}: xi+Rn > δ(xi+R)
⇔ δ < n·xn·xi+R
i+R
∀i∈ {j, k}: xi+Rn + y2 > δ(xi+R)
⇔ δ < n·xn·xi+R
i+R+2·(xy
i+R)
We can see that this boundary is not changed if it is not the least informed playern, who has the additional shared information. In a similar way we can also get the other boundaries, which result in five cases. In Figure 2.16 we can see how the shared information changes the duration. If the
0 | δ1
n·xn+R n·(xn+R)
|
2·x
n−1 2·xn−1+R
T1
−1
1 period 2 periods T periods
(a) 1. Case:xn6=xk, xn−16=xk
0 | δ1
n·xn+R n·(xn+R)
|
min 2·x
n−1+y 2·xn−1+R
T1
−1
, 2·x
n−2 2·xn−2+R
T1
−1
1 period 2 periods T periods
(b) 2. Case:xn−1=xk, xn−26=xj
0 | δ1
n·xn+R n·(xn+R)
|
min 2·x
n−1+y 2·xn−1+R
T1
−1
, 2·x
n−3 2·xn−3+R
T1
−1
1 period 2 periods T periods
(c) 3. Case:xn−1=xk, xn−2=xj
0 | δ1
minh
n·xn+R
n·(xn+R)+2·(xy
n+R),n·(xn·xn−1+R
n−1+R)
i
|
2·x
n−1 2·xn−1+R
T1
−1
1 period 2 periods T periods
(d) 4. Case:xn=xk, xn−16=xj
0 | δ1
minh
n·xn+R n·(xn+R)+2·(xy
n+R),n·(xn·xn−2+R
n−2+R)
i
|
min 2·x
n−1+2y 2·xn−1+R+y
T1
−1
, 2·x
n−2 2·xn−2+R
T1
−1
1 period 2 periods T periods
(e) 5. Case:xn=xk, xn−1=xj
Figure 2.16: Duration in case of shared information
79 shared information is split between two players that are both not the least or second least informed player, there is no change at all. This benchmark is shown in the first case.
In the second case the second least informed player has the shared information. This increases the upper boundary of the interval with a duration of two periods. So there exists an interval in which the information is centralized in two periods, while in the benchmark it takes T peri-ods. The boundary does not only depend onxn−1, but also on xn−2. Even in the benchmark δ <
2·xn−2
2·xn−2+R
T−11
has to hold, but as we described in Section 2.2, this is always satisfied for δ <
2·xn−1
2·xn−1+R
T−11 .
In the third case the third least informed player also has the shared information, so the boundary between a duration of2andTperiods depends onxn−3. This condition ensures that the players1 ton−3prefer to hold and will not deviate. In some cases the boundary between2andT periods will be the same for the second and third case, there is also the possibility that the boundary is higher in the third case.
The change in the fourth case is different. In that case the least informed player has shared infor-mation, while the second least informed player does not. This shifts up the upper bound for the interval in which the duration is one period. Both boundaries increase if the least and the second least informed player share the information. In that case the increase of the boundary between2 andT periods is stronger than in the second case. Whether the increase in the third or fifth case is stronger depends on several variables. The first boundary of the fifth case can also be larger than that of the fourth case.
We can see that as soon as at least one of the playersn and n−1 has shared information, the boundaries change. We still need the conditions to hold for all other players, so we use the mini-mum of both conditions.
These changes are different than those we did in Section 2.2.4. If we just increase xn orxn−1, we observe a different change. A comparison of these changes shows that for some values of the parameters the duration decreases more if the players have shared information, while for other values the duration decreases stronger if just one of the players has additional information.