periments
One way neural measures might be especially useful for economics is if they can add predictive power to other types of variables. In our study, a candidate prediction of a causal change comes from the well-studied ACC. It is known that scarce cingulate resources can be tied up if subjects have to perform a parallel activity which requires attention or working memory (e.g., remembering a several-digit number). Suppose sellers also have to remember a number while they are deciding
Figure 3.8: Both the left and right dorsal striatum are negatively correlated with subject level “Information Revelation” coefficient (the slope of the regression ofsonv), which we use as a proxy for buyer honesty.
what price to choose. If ACC resources are substituted away from the pricing decision and toward number-memory, then the negative correlation between ACC activity and information sensitivity suggests sellers will become more information sensitive (i.e., they will respond with prices more closely linked tos). That is, adding a number-memory treatment could change the outcome of the bargaining.
Another candidate prediction comes from the positive correlation between right DLPFC activity and higher price markups (seller prices above buyer-suggested prices). As described above, Knoch et al. (2006) [11] successfully used TMS to disrupt brain activity in right DLPFC and induce players in ultimatum games to accept low offers more frequently. Their interpretation is that the DLPFC is activated by a conflict between the desire to accept money and the desire to enforce social norms by rejecting a low offer. TMS disruption disables this region so behavior reverts to the simpler, more automatic reaction of acceptance. Similarly, one prediction from the DLPFC-markup link we see is that if TMS was used to disrupt activity in the right DLPFC in our task, markups would be reduced.
We don’t know if either of these experiments would have the predicted effect, or whether they are worth doing at all. We offer these suggestions, at this exploratory point, mostly as illustrations of the recipe of using the fMRI to identify regions of activity which can then be influenced by other treatments in a way that potentially changes behavior.
3.7
Conclusion
The behavioral data from this experiment show that subjects have a strong tendency toward both information revelation and information sensitivity, even in situations where economic theory predicts there should be none. The neural data linked to these behaviors suggest that rather than a social or emotional explanation (subjects want to be fair), this information transfer may be the result of a far more basic automated response to information. In fact, some activations seen in this study are very close to those seen in the classical information processing experiment, the Stroop task. These sorts of data suggest that we should reconsider what the “default” response of a decision maker might be to any piece of information.
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Appendix C
Bayes-Nash equilibria of the
one-shot game
Claim C.0.1 The only Bayes-Nash equilibrium of the one-shot game is one where the buyer’s sug-
gestion is uninformative.
Intuitively this follows from the fact that in this game the buyer and seller’s incentives are almost exactly opposed. They both prefer that a trade take place, but given that a trade does take place the game is constant sum. To prove this formally we first need to define a few functions.
The probability mass functions for both player’s strategy based on the information they re- ceive. For buyers this is the perfect information v, while for sellers this is the noisy and possibly uninformative suggestions.
s(x|v) = P r{suggestion=x|v}
p(x|s) = P r{price=x|s}
q(v|x) = Ps(x|v)
ys(x|y)
The utility functions for the buyer and seller respectively are:
UB(s, v) = X x<v p(x, s)ub(v−x) US(p, s) = us(p)X v≥p q(v|s)
Letn be the lowest price chosen by the seller with positive probability over all suggestions and chooses∗ such that p(n|s∗)> 0. Let S
0 be the set of suggestions such that p(n|s)>0. We will
show that any buyer with valuev > nwill always choose a suggested price inS0by induction. First
note that since n is the lowest price charged given any suggested price, suggestions inS0 strictly
dominate other strategies for buyers with valuen+ 1 since any other price yields a payoff of 0. Assume that all buyers with values in [n+ 1, m−1] all only choose suggested prices in S0.
So any s0 6∈ S
0 q(v|s0) = 0 for all values between n+ 1 and m−1, which in turn implies that
P r(v ≥p|s0) =P r(v≥m) for allv∈[n+ 1, m−1]. In other words, givens0,mdominates all prices in [n+ 1, m−1]. By assumptionp(x, s0) = 0 for allx ≤n so we have p(x, s0) = 0 for allx < m. ThereforeUB(s0, m) = 0 for anys0 6∈S
0 and buyers with valuem will always chooses∈S0. Note
that this means that all buyers with valuev < nmust always chooses∈S0 as well since otherwise
there would be somes6∈S0 such thatP r(v < n|s) = 1 implying thaty(p|s)>0 for somep < n. Lemma C.0.1 In equilibrium sellers choose prices according to some fixed distribution p∗(x) re- gardless of the suggested price received.
We show this using an induction argument like the one above. We need to show that p(x|s1) =
p(x|s2) =p∗(x) for all s1, s2 ∈ S0 that are sent with positive probability. To see this note that it
must be true forx =n since if there is anys1, s2 such thatp(n|s1)> p(n|s2)s1 dominatess2 for
buyers with valuen+ 1. Once again, this implies that s(s2|n+ 1) = 0 ⇒p(n+ 1|s2) = 0 and s1
dominatess2 for buyers with value n+ 2. Itterating up we see that s1 must dominate s2 for all
buyers ands(s2|v) = 0 for all v. Once again, using induction we see thatp(x|s1)∼=p(x|s2) for all
s1 s2sent with positive probability. In other words, any suggested price actually sent by the buyer
in equilibrium has no effect on the seller’s pricing strategy. LetS1 be the set of suggestions sent in
equilibrium.
Let x be some price sucht that p∗(x) >0. This means that x is a best response to s∗ ∈ S
1,
specifically it must be at least as good asx+ 1, andx−1 so
us(x) X v≥x q(v|s∗) ≥ u s(x+ 1) X v≥x+1 q(v|s∗) ⇒ us(x)q(x|s∗) ≥ (us(x+ 1)−us(x)) X v≥x+1 q(v|s∗) ⇒ us(x)s(s∗|x) ≥ (us(x+ 1)−us(x)) X v≥x+1 s(s∗|v)
Sincep∗(x) is the same for alls∗∈S
1we can sum over all the sinS1 and get
us(x) X s∗∈S1 s(s∗|x)≥(u s(x+ 1)−us(x)) X v≥x+1 X s∗∈S1 s(s∗|x).
We know that buyers will always chooses∗∈S
1, so this is simplified to
us(x)
us(x+ 1)−us(x)
≥10−x.
Sinceu2in increasing and concave the left side of this inequality is strictly increasing while the right
side is strictly decreasing. Also notice that if the inequality isstrict p∗(x+ 1) = 0, so the seller will mix between at most 2 pricesx andx+ 1 (this occuring only when the intersection pointx∗, such that 10−x∗= us(x∗)
us(x∗+1)−us(x∗) is an integer). This in turn implies that the seller simply choosesx