For games utilizing each of the aggregation mechanisms we considered, we showed that equilibrium messaging distributions are not representative of the underlying value distributions (rejection of the sincere conjecture). Indeed, for each type of message aggregation, we found classes of equilibria in which it is always the case that participation increases in the extremity of how an individual feels
about a product (confirmation of the extreme participation conjecture). We prove each of these with a unique model that includes rating aggregation; costly, optional participation; diffuse but correlated valuations; and intrinsic rating motivation. No other work has examined this topic while considering each of these facets simultaneously.
In the case of mean message aggregation, we showed that so long as messaging strategies are continuously differentiable, this pattern of participation must occur inallenvironments. We remain agnostic about the exaggeration conjecture, that raters skew their ratings towards the extreme. How- ever example 1 and proposition 2.6.6 provided classes of games in which the exaggeration conjecture is rejected.
Similarly, in the case of median message aggregation we examined a class of environments and equilibria in which we can reject the sincere conjecture and accept the extreme participation con- jecture. Unique to this message aggregator, we show that there are each of: equilibria in which the symmetric messaging strategy is to dampen players’ values; equilibria in which the symmetric messaging strategy is to exaggerate players’ values; and equilibria in which the symmetric messaging strategy is the sincere messaging strategy. This is because with the median aggregation mechanism, the marginal effect of a player’s message is either 1 or 0 (a message is either the median or it is not; if it is not, local changes do not affect the median value) and this implies that there exist many messages over which each player is indifferent.
In the case of a binary “thumbs up” or “thumbs down” mechanism, we showed that both senders and receivers utilize a cutoff strategy in equilibrium. We showed that in all equilibria, participation is increasing in the extremism of a sender’s value on at least part of the value domain, and we identify conditions under which the extreme participation conjecture holds over the entire domain.
Similarly to the other aggregation mechanisms, we showed that such equilibria exist in symmetric environments.
The selective participation result is highly reminiscent of the “Swing Voter’s Curse” (Feddersen and Pesendorfer 1996) in which voters who are uninformed about which candidate is best prefer to abstain in some equilibria. In contrast to that model, actors in our environment receive signals
that result in a continuum of posterior beliefs, and therefore there exists the possibility that in equilibrium players with different values participate at different (interior) rates. The participation result is therefore also partially due to the assumption that rating is idiosyncratically costly, and that senders’ costs are not related to their values. Restated, in equilibrium, players with stronger posteriors are more willing to pay to rate a product; therefore, since there exist a continuum of costs, we can say that players with extreme values are more likely to rate the product. If instead rating was costless or beneficial to all consumers, then we would not expect the hypothesized participation behavior to persist. Instead, we would observe full participation. However, full participation is plainly not observed.
The other factor behind the extreme participation result is that we were able to identify equilibria in which the marginal effect of equilibrium messages, as a function of senders’ values, exhibited the same comparative static as did posterior strength. We achieved this by narrowing the focus to equilibria in which the receiver played a cutoff strategy. Such a consideration was not a factor in the swing voter literature since both signals and actions in that model were binary.
It is through this vector that the choice of aggregation mechanism affects outcomes. Clearly, aggregation does not affect the informativeness of signals. It does affect the way messages are incorporated with each other, and therefore it does effect the marginal effect of any given message.
It should be noted that with rational agents, message aggregation is weakly welfare reducing. An aggregation mechanism cannot do better than allowing the receiver to view each individual rating because the receiver can best infer the state of the world by observing senders’ signals directly. A proper treatment of the welfare effects of message aggregation is therefore likely to include agents with limited capabilities to consume or update properly given a multitude of signals. This is a potential direction for future work.
−1 0 1 1
V
fv|θl(v) fv|θh(v)
a) Densities of values, conditional on the state of the world
−1 0 1
1
V
P r(θh|v1) P r(θl|v1)
b) Relative Bayesian posterior beliefs
Figure 2.1: Example conditional value densities and posterior beliefs
−1 1
0 0
1
X(v1) P(v1, c1)
c1=18
V
a) No messages sent implying preference for the risky product (µΦ ∈ H) results in a one-sided cutoff participation strategy
−1 1
0 0
1
X(v1)
P(v1, c1)
c1=18
V
b) No messages sent implying indifference (µΦ ∈ I) results in a two-sided cutoff participation strategy
Figure 2.2: Example equilibrium participation strategies with one sender whenc1= 0.2
−1 0 1
ˆ
µ(M1∗(vA1)) ˆ
µ(vA1) ˆ
µ(v1B) ˆ
µ(M1∗(v1B))
fµ(m−1)|θh(tr) fµ(m−1)|θl(tr)
a) Full participation,N= 3, densities of aggregate messages when excludingm1 and conditional on the state of the world
−1 0 1
1
M1∗(vA1)
v1A v1B M1∗(vB1) P r(θh|v1)
P r(θl|v1)
fµ(m
−1 )|θl( ˆµ(m1)) fµ(m
−1 )|θh( ˆµ(m1))
b) Full participation,N = 3, relative marginal utility ofm1, compared to the sender’s relative posteriors as a function ofv1
Figure 2.3: Example best response messages for example valuesvA1 andv1B, given full participation
−1 1
0 1
−1
v1A v1B
M1(v1) =v1
M1∗(v1)
V
a) Sender 1’s best response, given the assumed strategies
−1 1
0 1
−1
vA1 vB1
M(vi) =vi
M∗(vi) V
b) A symmetric equilibrium messaging strat- egy
Figure 2.4: Example Messaging Under Full Participation
−1 0 1
1 8
X(v1)
V
a) Example equilibrium net expected utility to participation as a function of a player’s value
−1 0 1
1
m2 fS2|θl(m2)
fM2|θl(m2)
b) Densities of sent messages under full par- ticipation and voluntary participation when M2(v2) =v2
Figure 2.5: Example voluntary participation and its effect on message likelihood
−1 0 1
ˆ µ(vA1) ˆ
µ(v1B)
fµ(m−1)|θh(tr) fµ(m
−1)|θl(tr)
a) Voluntary participation,|n|= 2, densities of aggre- gate messages when excludingm1 and conditional on the state of the world
−1 0 1
1
M1∗(v1A) =vA1 M1∗(v1B) =v1B P r(θh|v1) P r(θl|v1)
fµ(m
−1 )|θl( ˆµ(m1)) fµ(m
−1 )|θh( ˆµ(m1))
b) Voluntary participation,|n|= 3, relative marginal utility of m1, compared to the sender’s relative posterior as a function ofv1
−1 0
1
ˆ
µ(M1∗(vA1)) ˆ
µ(vA1) ˆ
µ(vB1) ˆ
µ(M1∗(vB1))
fµ(m−1)|θh(tr)
fµ(m−1)|θl(tr)
c) Voluntary participation,|n|= 3, densities of aggre- gate messages when excludingm1 and conditional on the state of the world
−1 0 1
1
M1∗(vA1)
vA1 vB1
M1∗(vB1) P r(θh|v1)
P r(θl|v1)
fµ(m
−1 )|θl( ˆµ(m1)) fµ(m
−1 )|θh( ˆµ(m1))
d) Voluntary participation,|n|= 2, relative marginal utility of m1, compared to the sender’s relative posterior as a function ofv1
Figure 2.6: Example best responses for example valuesv1A andv1B as a function of level of partici- pation, given the assumed strategies of other players
−1 1
0 1
−1
vA1 v1B
M(vi) =vi
M∗(vi) V
Figure 2.7: Example symmetric equilibrium messaging strategy, given voluntary participation