Results: Play in the Ultimatum Game

generous offers becomes stronger and it does so especially for relatively less generous offers. This explains why as one considers wider cost dis-tributions receivers’ probabilities to deliberate conditional on receiving an offer p ≥ 5 become more heterogeneous.

Finally, consider the case in which a receiver is offered an amount p = 0. As Figure 17 shows this is the scenario in which receivers de-liberate less frequently. This is likely the combination of no incentive to deliberate (as in the case of offers p ≥ 5) together with stronger selection pressure. The low incentive to deliberate arises because independently of the actual game played receivers are in expectation better off by reject-ing a null offer. In fact, acceptreject-ing the offer implies a null payoff, while rejecting the offer implies an expected payoff of βπ – or β(︁1 − E[C(kp)])︁π if proposers adopt a differentiated strategy. Instead, the higher selection pressure comes from the fact that βπ is always smaller than any generous offer considered before. This can also explain why receivers’ probability to deliberate conditional on being offered an amount p = 0 is increasing in β: for low values of β receivers expected payoff is small and so selec-tion pressure on deliberaselec-tion patterns is strong while for high values of βreceivers’ expected payoff converges to π = 4.44 which can be consid-ered as a ’generous offer’ and, consequently, (i) the strength of selection is low and (ii) receivers probability to deliberate gets close to the ones of generous offers.

ac-count simultaneously proposers’ and receivers’ expected behaviors. In other words, these are the rejections rates one should observe in an UG according to the model.

I begin by presenting the results related to conditional rejection rates.

Conditional rejection rates are computed as follows. For any given (β, C) combination I compute the average receivers’ rejection rate conditional on receiving an offer p both under intuition and under deliberation if the game is an UG. Then, for any possible offer p, I derive the probability that a receiver plays intuitively or according to deliberation given that it has been offered an amount p. Finally, I use this information regarding receivers’ probability to deliberate as weights for intuitive and delibera-tive rejection rates, thus obtaining conditional rejection rates.

In Figure 18 I report receivers’ conditional rejection rates by offer p as a function of β for each cost distribution considered.

Overall conditional rejection rates are decreasing in the offer p: by making a higher offer a proposer usually guarantees himself a higher chance of having its offer accepted. However, there are some exceptions to this general rule. For example, in the case C = 1 for many intermediate values of β if a proposer offers four rather than three then it faces a higher rejection rate. I explain this in the light of Figure 17 which describes receivers’ deliberation patterns conditional on the offer received. In fact, by offering four rather than three a proposer triggers lower deliberation on the receivers’ side which in turn increases receivers’ rejection rates as rejection rates under intuition tend to be higher than the ones under deliberation if the game is an UG.

As in the case of receivers’ probability to deliberate given an offer p I now distinguish rejection rates conditional on offers p ∈ {1, 2, 3, 4} from the ones conditional on null offers and the ones conditional on generous offers p ≥ 5.

As the figure shows, receivers’ rejection rates conditional on being of-fered an amount p ∈ {1, 2, 3, 4} are weakly increasing (non-decreasing) in βfor a given cost distribution. More precisely, they tend to be stable and close to zero for low values of β but once β reaches a critical level (which depends both on p and C) conditional rejection rates increase

dramati-Figure 18: Conditional rejection rates. Average receivers’ conditional re-jection rates in the Ultimatum Game and their standard errors as a function of β by cost distribution.

cally until they converge to a maximum level. I explain this pattern as follows. For low values of β receivers tend to accept any offer both under intuition and under deliberation if the game is an UG and, consequently, conditional rejection rates are close to zero for low values of β. However, as β increases receivers begin to reject low offers under intuition (in or-der, offers equal to one, two, three). After receivers start to reject offer p under intuition, rejection rates conditional on offer p remain low as long as receivers deliberate frequently enough conditional on being offered an amount p; when they stop doing so rejection rates rise steadily. This explains why conditional rejection rates begin to increase after β reaches a critical value that depends both on p and C.

It is interesting to underline that conditional rejections rates reach a

maximum level that is increasing in C and also depends on p. This is because the maximum level reached by conditional rejection rates is de-termined by the probability that receivers incur into deliberation given that they have been offered an amount p. More precisely, if receivers de-liberate a fraction of times x then conditional rejection rates can be up to 1−xas under deliberation receivers always accept the offer if the game is an UG. This together with the findings that deliberation levels are lower in the case of wide cost distributions and they also depend on p – as seen in Section 4.3.2 – explains why the maximum level of conditional rejec-tion rates is higher in the case of wide cost distriburejec-tions and also depends on the offer p.

Conditional rejection rates of fair offers p ≥ 5 are always close to zero, as expected, given that both under intuition and under deliberation if the game is an UG receivers always accept these kinds of offers. Instead, re-jection rates conditional on null offers are always high, but they are never equal to one. This holds because it is always the case that under intuition receivers reject null offers while under deliberation they accept zero of-fers with high probability. Despite these opposing effects the first one (intuitive rejection) dominates as receivers deliberate with low probabil-ity if they are offered nothing. Interestingly, the shape of the conditional rejection rates associated with null offers closely remembers receivers’

probability to deliberate conditional on p = 0 inverted.

It is worth underlining that the model can account for high condi-tional rejection rates in the UG for offers up to 40% of the value to split.

I move now to the analysis of overall rejection rates implied by the model. These rejection rates have been computed as follows. For any (β, C) combination I have derived proposers’ expected probability to make any possible offer under intuition and under deliberation if the game is an UG; moreover, I have computed proposers’ probability to incur into deliberation. With these measures at hand, I have derived the probabilities that proposers make any given offer in the UG. Then, I com-bined these measures with the corresponding conditional rejection rates to derive a measure of overall rejection rates. The rejection rates thus obtained are illustrated in Figure 19.

Figure 19: Expected rejection rates. Overall expected rejection rates in the Ultimatum Game and their standard errors as a function of β by cost dis-tribution. These rejection rates are computed by taking into account both proposers’ behavior and receivers’ conditional rejection rates.

As the figure shows, expected rejection rates in the UG are overall in-creasing in the probability that the game is a simplified BG. This might be expected as the higher β the more receivers are demanding and, thus, the more they should reject offers. However, interestingly rejection rates in the UG are not monotonically increasing in β. In fact, expected rejec-tion rates present multiple peaks and troughs. By comparing this finding with Figure 14 it seems that implied rejection rates decrease at values of β in correspondence of which proposers’ average offer switches from one value (say p^′) to the next available one (p^′+ 1).

There is no clear relationship between expected rejection rates in the UG and the cost distribution considered. However, rejection rates are slightly lower in the case of narrow cost distributions. Moreover,

un-der narrower distributions expected rejection rates are more stable and monotone.

Overall, in the absence of players’ mistakes in implementing their strategy, the model predicts rejection rates between 5% and 10%. These figures are significantly lower than the rejection rates observed in the laboratory but still can explain a relevant part of the phenomenon.

Appendix G and Appendix H report some robustness checks. The former analyses the role played by the shape of the cost distribution, while the latter studies the effects associated with changes in agents’ pa-tience factor.