Identificación de las soluciones propuestas por ASF

When YouTube abandoned its user generated rating system, it replaced it with a system that prompted users to rate a video as “thumbs up” or as “thumbs down”. Binary-message mechanisms, such as this, can be found in many implementations, and are therefore examined here.

To examine the “thumbs up, thumbs down” (TUTD) environment, we must first define a binary message space. Thus, let each sender’s message space be {m, m}, where m and m are arbitrary but unique messages. Next, we define the TUTD message aggregator, µt_{, to display the number} of senders who send m and the number of senders who send m; that is, the receiver’s type space becomes{0,1, ..., N} × {0,1, ..., N}. The receiver’s inference problem is thus vastly different than in the previous treatments. Most significantly, the receiver can now see precisely which messages are sent, and precisely the level of participation. This will result in the receiver playing a cutoff rule for each level of participation. For the senders, despite best responding to a potentially more complex strategy on the part of the receiver, the best response strategy is simple.

Lemma 2.6.10. In any TUTD game, the equilibrium messaging strategy is a cutoff strategy.

Lemma 2.6.10 simply says that there exists some value below which senders send one message, and above which, they send the other. We call this switch-pointvt, and for ease of exposition (and without loss of generality) we assume that players sendmfor values belowvt_.

M(vi) =        m, ifvi< vt m, ifvi> vt

This simple message strategy results in a simple density of sent messages.

P r(m|θ) =        Rvt v fv|θ(v)Fc(X(v))dv, ifm=m Rv vtfv|θ(v)Fc(X(v))dv, ifm=m

By our assumptions on the conditional distributions of values, it is quick to see that P r(m|θl_)>

P r(m|θh_{) andP r(m|θ}l_{)< P r(m|θ}h_{). That is, players are more likely to send the low message in} the low state than they are in the high state, and players are more likely to send the high message in the high state than they are in the low state. As mentioned above, this results in the receiver playing a set of contingent cutoff strategies.

Lemma 2.6.11. In any TUTD game, in equilibrium, conditional on total participation levels, the receiver plays a cutoff strategy in the difference between the number of low messages received and the number of high messages received.

Lemma 2.6.11 says, quite intuitively, that for a given number of total messages, as the number of high messages becomes greater, and the number of low messages becomes fewer, the receiver’s posterior belief that the state of the world is the high state, increases.

Proposition 2.6.12. In any TUTD game, in equilibrium, at least one of the following is true: X(vi)is decreasing forvi< vt; andX(vi)is increasing forvi> vt.

Thus, in all equilibria of the class of TUTD game, participation patterns resemble the extreme participation conjecture on at least part of the domain. Corollary 2.6.12 gives a condition under which the conjecture holds on the entire domain.

Corollary 2.6.12: There exists an open neighborhood around vI_{, V}0_{, such that in equilibrium, if} vt_∈V0_{, thenX(v}

i)is decreasing for vi< vt and increasing forvi> vt.

The issue here is that the messages m and m are both generated by a wide range of values. Corollary 2.6.12 uses the fact that if the cutoff,vt, is sufficiently close tovI, then the set of values

that generate each message are similar to one another.

Restricting the set of environments we are considering to the simple symmetric environment that we established with example 1 and proposition 2.6.6, we see that the requirement of corollary 2.6.12 forvt_{to be nearv}I _{is not unreasonable.}

Proposition 2.6.13. For any game with an environment such thatv=−v andfv|h(v) =f_v|l(−v), there exists an equilibrium in whichvt_=vI _{= 0, and thereforeX(v}

i)is strictly decreasing forvi<0

and strictly increasing for vi>0.

Focusing on YouTube’s newest implementation, users cannot see ratings before they begin to view a video. This puts YouTube’s implementation outside of the scope of the TUTD analysis, but not outside the scope of our model. Our predicted comparative statics depend on the environment, but if we assume the symmetric environment and an equilibrium of the type described by proposition 2.6.13, we find that X(vi) is positive everywhere but atvΦ. In YouTube’s implementation, since there is no interaction between senders and receivers, X(vi) would be 0 for all vi. That is, our model predicts that under their new implementation only those consumers who enjoy the act of rating (negativeci) will submit ratings. This implies that we would predict that YouTube’s change of mechanism decreased ratings for all values, excluding some central value. Of course, since there are but two messages in the TUTD system, the only testable prediction is that participation ought to have decreased in aggregate.