Predictions

A theoretical analysis of the one shot game shows that if a seller’s strategy is at all sensitive to suggested prices, high-value buyers will all pool with the low-value buyers, transmiting minimal information and making the seller’s strategy non-optimal (see appendix for formal details). So the equilibrium prediction is that suggested price should be completely unrelated to value (P rob(v|s) = .1 for allv, s), and the seller will choosepmaximize u(p)(11−p)/10, for a risk-neutral seller this implies that he should always choosep= 5 orp= 6.

Keep in mind that in the theory above, subjects are formingcorrect beliefs about what other players will do and maximizing their own personal gains given their beliefs. As a useful benchmark contrast, suppose both players were trying hard to cooperate and earn the most money together from the task (e.g., suppose they had planned beforehand to split their earnings equally, so they want to maximize the total gain). Since the seller always wants to sell the good to maximize their joint gains, the seller should try to always name a price the buyer can afford. One way to do this is to always name a price of 1. Another way to do this is for the buyer to make a suggestion which is communicates the value perfectly, i.e., the suggestion correspondence has a functional inverse f(s) =v, and for the seller to choosep≤f(s). Such patterns could emerge, in theory, but neither emerged in this experiment, instead we observe partial information revelation of the sort you might expect if the buyer and seller incentives were somewhat aligned.

While subjects were not given any feedback about the outcome of each trial, both the buyer and seller are aware of the buyer’s history of suggestions over the course of the experiment. This raises the question, do sellers pay attention to the buyer’s “reputation” and, if they do, do buyers anticipate this by revealing information. We consider two ways in which the history of suggestions might create repeated game effects. First, we consider purely strategic models where sellers attempt to glean information from not only the current suggestion, but the entire history of suggestion. Second, we will look at non-standard preferences that include a taste for retaliation or fairness.

One simple way to model information extraction from a buyer’s history is to look at an extension of the cognitive hierarchy model [4]. Assume that level-0 buyers use a strategy of the form s = max(1,bαvc), withαdrawn randomly from [0,1]. Higher-level sellers assume that any given level-0 player uses the same α throughout the experiment, so they can infer their opponents type from

the suggestions they send. Level-nbuyers choose signals to maximize their sum payments for the

current period and thenfollowing period (sellers do not consider future payoffs since their actions,

unlike buyer actions, will not affect buyer beliefs in the future).

First let us consider the level-1 sellers. Before the first round they have a uniform prior on the

valueαthat their opponent uses to generate suggestions. At the beginning of the first round they

will receive a signal from the buyer, s. The seller can update their prior on the value of αusing

this signal and Bayes Law. For example, ifs= 8, the seller will update their perceived probability

distribution onαso that they believe thatP(α < .8) = 0. Sellers then best respond tosgiven their new updated prior. In each subsequent round the seller will update their prior again based on the new signal. Here we can make two different assumptions. First, the sellers may use the entire history of signals to generate a new prior. In this case if the seller ever sees a particularly high signal, like

s= 1, beliefs will always be such thatαmust be high and sellers will always be credulous. Second,

sellers may have a limited memory, and thus only update the original uniform prior on the value of

αusing the current signals0and the signal from the previous rounds−1. Here, the effects of a high

signal are transient.

Level-1 buyer behavior is uninteresting. Since they believe that the sellers are credulous, they will

always choose a signals= 1. However, Level-2 buyers will anticipate the fact that their cheap talk

signalswill affect the seller’s behavior. When the buyer believes that seller memory is unlimited,

they can build up their credibility, i.e., manipulate the sellers to believe that α= 1, but signaling

10 just once. They will signals= 1 in all subsequent rounds. If, on the other hand, buyers believe

that sellers have a limited memory, there will be an inverse relationship between the buyer’s valuev

and the signal chosens. In essence, buyers build credibility when they have low values by sending

high signals in order to gain more surplus when they have high values. The average best response

by a buyer given their previous signals−1 and current valuev-are shown in Figure 2 below.

A model where sellers have retaliatory preferences, i.e., they enjoy punishing buyers who send low signals, yields buyer behavior similar to CH level-2 buyers since sellers never know if their punishment is justified. Buyers will still use rounds where they have low values, and therefore little potential payoff, to raise their average signal and fend off retaliation by the seller. When they have higher values, they will send low signals to get better prices.

2 4 6 8 10 1 ₂ 3 ₄ 5 ₆ 7 ₈ 9 ₁₀ 2 4 6 8 10 s₋₁ v Optimal Signal 1 2 3 4 5 6 7 8

Figure 3.2: The average best cheap talk signal (sometimes the best response set has multiple values) for a level-2 buyer givenτ = 1.5 over all possible valuesv, and previous signals,s−1, when the seller

has a limited memory

Model Predicted Buyer Behavior

Predicted Seller Behavior Nash Babbling p= 5 or 6

Cognitive Hierarchy with full memory level-0 s= max(1,bαvc) p=s

level-1 s= 1 for allv argmaxp(p·P r(v≥p|{st})) where{st}is the series of signals

until the current time level-2 s= 10 early in the

experiment,s= 1 from there on

argmaxp p· P r(l= 1|{st})11−₁₀p+P r(l= 0)P r(v≥p|{st})

Cognitive Hierarchy with limited memory

level-1 s= 1 argmaxp(p·P r(v≥p|s0, s−1))

level-2 s = 10 for low v,

s= 1 otherwise

Similar to above but priors depend only on recent information.

0 1 2 3 4 5 6 7 8 9 10 0 2 4 6 8 10 Signal

Signaling Data and regression for AI−2, slope = .6, intercept = .4, R2 = .84

Value 0 1 2 3 4 5 6 7 8 9 10 0 2 4 6 8 10 Signal Price

Pricing Data and regression for AI−2, slope = .92, intercept = 2.75, R2 = .61

Figure 3.3: Suggested prices sent by a single subject as buyer, and prices set as seller. (Points are jittered by adding random noise so that identical points are plotted separately).