the multiple-sender games, there are multiple possible equilibria of a one-sender game. However, in this restricted environment, the number of potential equilibria is significantly reduced.
Proposition 3.4.4. For every game, there exists a δ <0.5 such that in every equilibrium of the game, one of the following must hold: aΦ = 0; aΦ= 1; or aΦ ∈(0.5−δ,0.5 +δ). The final case may constitute part of an equilibrium if and only if the strategies of that equilibrium also form an equilibrium when aΦ= 0.5.
Proposition 3.4.4 says that there can only be an equilibrium in which the receiver mixes upon receipt of the null message if the strategies that make up that equilibrium are also an equilibrium when the receiver mixes evenly between the two products.
This result is particularly useful because, as discussed in Chapter 2,aΦhas a large effect on the form equilibrium strategies. Proposition 3.4.4 says that we may, without significant loss of generality, limit our search for equilibria to the three cases in whichaΦ∈ {0,0.5,1}.
for any learning effects. Each of the possible sequences was implemented one time, yielding six total sessions. Each treatment of each session lasted 30 matches, and as a result we conducted 180 total matches for each treatment.
Each session had 12 participants, each of whom were Caltech students. Subjects took part in one session each, and we therefore had 72 total, unique participants.
In each match subjects were randomly grouped into groups of size N + 1, where N was the number of senders in that treatment. N of the subjects in each group were randomly selected and assigned the role of sender. The remaining subject in each group was assigned the role of receiver.
The number of receivers in each group in each match did not change. After each match, groups and roles were re-assigned at random.
Instructions were read aloud to subjects while slides depicting important points in the instructions were displayed at the front of the room.4 Experiments were conducted via networked, private computer terminals, each separated by a physical divider. Subjects’ computers were coordinated by a central server computer, controlled by the experimenter, and which also was responsible for realizing values of random variables in each match.
Subjects were told that they were participating in an experiment evaluating product rating be- havior, and that they would need to discern between an old product of known quality, and a new product of unknown quality. This change in diction from “safe” to “old” and “risky” to “new” was done to avoid activating subjects’ risk preferences. The instructions and slides took the subjects through an example match step-by-step. Subjects were fully informed of the parameters and dis- tributions of the game; they knew the distribution of costs as well as the conditional densities of values. It was furthermore explained to them that the larger the value they received, the more likely it was that the state was the high state. Whenever an example random variable was realized in this step-by-step instructional process, an additional example draw was shown in which the example sub- ject got a different draw (in the opposite direction, if applicable). After the instructions concluded, subjects participated in 5 practice matches and were given an opportunity to ask questions before
4A copy of the instructions and slides for the session in which the treatment order was (N=2, N=5, N=1) can be found in Appendix D. The final three slides were revealed one bullet point at a time.
and after the practice matches. A summary slide of the important points of the game was left on display at the front of the room throughout the session.
For a given group, matches proceeded in the following way (all actions and random variables were independent across groups). First, the state of the world was determined by the server. Next, each sender received an i.i.d. draw from the appropriate distribution of the new/risky product (their signal). This draw was described to participants as a “clue about the quality of the new product”.
Each sender was asked which message from M they would prefer to send if they were to send a message. Senders were then each shown their private cost of sending their chosen message, and asked whether or not they wished to participate.5 The group’s receiver then viewed the average of all the messages that were sent. If no messages were sent, the receiver saw the message “No Messages Were Sent”. Finally, the receiver chose either the new or the old product. If he chose the old product, then he received 100 points. If he chose the new product, then he received a draw from the same conditional distribution of values from which the senders previously drew. Senders’
payouts for the match were equal to the receiver’s payouts for that match less whatever costs they may have incurred through messaging. Therefore, it was in each sender’s best interest to guide the receiver to the best choice.
During the instructions, the game was described in terms of the first treatment, and subjects were told that these conditions would last for 30 matches. Subjects were told that there would be two additional parts of the experiment after the initial 30 matches were completed, but were not immediately informed what those parts would entail. At the conclusion of the first treatment, the experimenter described the change in the number of senders for the second part of the experiment.
The summary slide at the front of the room was updated, subjects were allowed to ask questions, and then the second treatment proceeded as did the first. At the conclusion of the second part, this process was repeated. At the conclusion of the session each subject was paid in cash, in private, the sum of his or her earnings over all 90 matches. Subjects earned approximately $22, on average plus a $5 show-up payment, and each session lasted approximately one hour.
5The order of these events was meant to enforce the theoretical independence of message selection and costs.
Figures 3.2 and 3.3 display the unique equilibrium messaging and participation strategies, as a function of values, for senders in each treatment whenaΦ= 0 and whenaΦ= 0.5.6 For theN = 1 case, the displayed strategies are exact. For N >1, the equilibria were computed numerically, and this reflects the fact that the game is more complex with multiple senders. In each of the figures, the black dotted line corresponds to theN = 5 case, the red dashed line corresponds to theN= 2 case, and the blue solid line corresponds to theN = 1 case. In Figures 3.2b and 3.3b, for a given signal, multiple values of the same color indicate indifference between those values; when N = 1, senders are indifferent between which message they send, so long as it induces their desired action from the receiver. TheaΦ= 0 equilibria specify values for which there exist no cost low enough such that the sender’s best response is participation. For these types, the message specified in Figure 3.3b still represents the optimal message, conditional on sending a message. However, that information set will never be reached, in equilibrium.