The dynamics of behavior in modified dictator games

We investigate the dynamics of individual pro-social behavior over time. The dynamics are tested by running the same experiment with the same subjects at several points in time. To exclude learning and reputation building, we employ non-strategic decision tasks and a sequential prisoners-dilemma as a control treatment. In the first wave, pro-social concerns explain a high share of individual decisions. Pro-social decisions decrease over time, however. In the final wave, most decisions can be accounted for by assuming pure selfishness. Stable behavior in the sense that subjects stick to their decisions over time is observed predominantly for purely selfish subjects. We offer two explanation for our results: diminishing experimenter demand effects and moral self-licensing.

then a selfish individual will always maximize his own payoff. The implications for the definition of 'S-consistency' in our games are summarized in Table A. An S-consistent player A will always take the maximum possible amount from player B in the take games (i.e. will always choose option 11), will always transfer the minimum possible amount to player B in the give games (i.e. will always choose option 1), and will always choose d in PD games.
By definition, all types of other-regarding preferences include selfish preferences as a special case.

S1.2 Consistency according to inequality aversion preferences
Our second concept of consistency takes into account notions of inequality aversion.

Preferences of this type have been introduced by [1] and [2]; [3] also consider inequality
aversion, but additionally include reciprocity concerns in their preference model. Note that, in our games, considering reciprocity along the lines of [3] does not change the predictions for individual behavior. In particular, the specific concepts by [3] and by [1] would lead to the same results in our games: For a two-person game with players A and B, [1] specify individual A's preferences concerning his own payoff πA and his opponent's payoff πB: π < π π π α π π π π π β π = π , π u = U for for .
The corresponding specification by [3], p.822 reads where q is an indicator variable signaling the presence of reciprocity. In the case that the opponent, player A, "misbehaves" ([3], p. 822), q becomes 1  = q . As we designed our experiments in a way that avoids reciprocity, we can, for our paper, set 0  q , which leads to the simple specification This can be re-written as In the following, we present a more general notion of inequality aversion which is inspired by the three papers above. With preferences including inequality aversion, a player gains utility from his own payoff and loses utility from a difference between his own and the other's payoff. The utility function can be written as follows The implications for the definition of 'I-consistency' in our games can be derived in the following way: Given the utility function which characterizes inequality aversion, in the first case (1) it holds that and the marginal rate of substitu- In the second case (2) it holds that

Dictator games
Following our general notion of inequality aversion, in the give games, case (2) applies and the MRS is always larger than -pA (i.e. the slope of the budget line). Consequently, an optimum choice will always be represented by a corner solution. Given that player A profits from a higher πA and from a lower πB, an I-consistent player A should keep everything to himself: I-CONSISTENCY IN GIVE GAMES: An I-consistent player A in a give game will transfer no money to player B, i.e. γG1 = γG2 = γG3 = γG4 = 0.
Things are slightly more complicated if player A has a higher payoff than player B (πA ≥ πB), as is the case in our take games. Changes in πA will have two opposite effects. On the one hand, a rise in πA increases utility. On the other hand, this also increases the inequality between πA and πB, which, in turn, decreases utility. Given that behavior is best described by (1a), then three different notions of I-consistency in the take games apply. First, if the MRS is greater than -pA T1 , then player A should always take nothing. Second, if the MRS is less than -pA T4 , then player A should always take everything. Third, if the MRS is less than -pA T1 , but greater than -pA T4 , then player A should not take more from player B as the relative price pA of his own payoff increases. Note that these three notions require that preferences are homothetic. Given that behavior is best described by (1b), then I-consistency implies that player A always takes nothing since the MRS is greater than -pA T1 .
I-CONSISTENCY IN TAKE GAMES: An I-consistent player A in a take game will not take more from B as the relative price pA of his own payoff increases, i.e. τT1 ≥ τT2 ≥ τT3 ≥ τT4.

PD games
Given the previous remarks, for players A with inequality aversion preferences the definition of consistency in the prisoner's dilemma games is as follows:

S1.3 Consistency according to altruistic preferences
Our second way of modeling other-regarding preferences is in the spirit of the concept introduced by [4]. According to this concept, one's own and the other's payoff are considered as 'normal goods'. Assuming that player A derives non-negative utility from her own payoff and from player B's payoff and ruling out the case of simultaneous indifference with regard to both, the utility function can be written as follows: Using the additional assumption that neither πA nor πB are inferior, we conclude that 'demand' for πA and πB must each be falling in relation to its respective price. The implications for the definition of 'A-consistency' in our games can be characterized as follows:

Dictator games
Since πA and πB are normal goods, optimum demand for πB should not decrease with the relative price pA, i.e.
Consequently, in the take games, the amount taken from player B, τ = 500 -πB, should not increase with pA, i.e.   Given that in the four take games pA T1 < pA T2 < pA T3 < pA T4 , our definition of consistency in these games is: A-CONSISTENCY IN TAKE GAMES: An A-consistent player A in a take game will not decrease πB (will not take more from player B) as the relative price pA of his own payoff increases, i.e. τT1 ≥ τT2 ≥ τT3 ≥ τT4.
Accordingly, in the give games, the amount given to player B, γ = πB − 500, should not Given that in the four give games pA G1 > pA G2 > pA G3 > pA G4 , our definition of consistency in the give games is: A-CONSISTENCY IN GIVE GAMES: An A-consistent player A in a give game will not decrease πB (will not give less to player B) as the relative price pA of his own payoff increases, i.e. γG1 ≥ γG2 ≥ γG3 ≥ γG4. our games, I-consistency is a special case of A-consistency (and, accordingly, S-consistency is a special case of I-and A-consistency). All consistency definitions are summarized in Table   A.

Consistency within games
In order to investigate whether subjects behave consistently within any one of the three classes of games, we use the three concepts of consistency introduced in section 3. Note that, as already mentioned, these concepts are not independent measures. Since selfishness (Sconsistency) is a special case of I-consistency and I-consistency is a special case of Aconsistency in our games, the additional explanatory power of the latter concepts is limited to the cases where behavior is non-selfish or not I-consistent. Therefore, we should expect that most of the decisions are A-consistent and few are S-consistent. Fig A displays the results for all three waves:

Fig A. Frequency of consistent behavior in all games in the three waves.
For the take games, the concepts of I-and A-consistency are identical and explain 54 percent of all individual moves in the first wave, whereas 20 percent of the subjects display consistently selfish behavior. That is, 34 percent of observed decisions in the first wave can additionally be interpreted as consistent when applying notions of consistency based on inequality aversion or altruism. In the second wave, we observe a highly significant increase in consistent behavior. Now, more than 85 percent of decisions are I-and A-consistent. The observation that the difference between the frequencies of selfish and other-regarding choices remains the same implies that the increase in consistency is mainly caused by the increase in the share of consistent selfishness. In the third wave, the share of consistently selfish behavior again increases significantly, this time to 89 percent. Only the decisions made by the 11 per-0% 20% 40% 60% 80% 100% S-consistent I-consistent A-consistent cent of subjects who do not take all the money from players B can be accounted for by I-and A-consistency only. Remarkably, we do not observe any individual move in wave 3 which cannot be characterized as consistent in the sense of one of the three concepts.
In the give games, three quarters of the subjects already behave consistently selfishly in the first wave. This share significantly increases from wave 1 to wave 2, but does not significantly change from wave 2 to wave 3. S-consistency alone explains about 85 percent of the individual behavior observed in wave 3; there is no significant additional explanatory power offered by the other two concepts of consistency. Since those who decide to give something to player B choose, averaged over all of their decisions, only very small amounts, it seems to be adequate to characterize overall behavior in the give games as "consistently selfish".
In the PD games, we find different patterns of behavior. All the subjects behave in an A-consistent manner in the first wave. The reason for the good performance of A-consistency is the fact that all players A follow one of the four A-consistent patterns: they always defect (for the latter this is only true in wave 2). The reason is that strategies that are not S-consistent ('tit-for-tat', 'inverted tit-for-tat', and 'always c') are hardly ever used any more. We summarize in observation S1-1: Observation S1-1 The share of consistent moves within the three classes of games increases during the course of the experiment. This increase is almost always due to the fact that, over time, more and more subjects make consistently selfish decisions.

Consistency across games
The strongest test for the consistency concepts of individual behavior is the comparison of decisions made by any single subject in different games. Fig B summarizes our findings concerning the consistency over all three classes of games. We employ the definition of across-game consistency introduced in section S1.3 in all three cases.

Fig B. Consistency across games.
In the first wave, only 11 percent of all the subjects exhibit consistently selfish behavior in all ten games. There are significantly more decisions that are I-consistent than S-consistent, though not significantly more that are A-consistent than I-consistent. A-consistency, the most general concept of consistency, accounts for the decisions made by 46 percent of the subjects.
The latter observation implies that, in wave 1, more than 50 percent of all subjects behave inconsistently over the three classes of games. In wave 2, overall inconsistency decreases to 31 percent, which is largely due to a significant increase in the frequency of S-consistent behavior from wave 1 to wave 2. Again, there is no significant difference between the number of I-and of A-consistent subjects. The shares of S-, I-, and A-consistent decisions do not significantly change from wave 2 to wave 3. We summarize in observation S1-2: Observation S1-2 While in the first wave consistent selfishness is rarely observed across games, it dominates behavior in the second and the third waves. In the last wave, about three quarters of all decisions can be characterized as consistent across games.
Combining the observations on the behavioral dynamics with our three concepts of consistency reveals that there are only five out of 35 subjects who behave consistently across all games and do not change their behavior over the three waves. Four of these subjects behave strictly selfishly. Five subjects change their behavior from I-consistent to S-consistent and two subjects change their behavior from A-consistent to S-consistent over the three waves.
That is, all consistent changes over the three waves are from less to more selfishness. In the last wave, 11 out of the 12 subjects revealing consistent decisions in all three waves can be 0% 20% 40% 60% 80% 100% wave 1 wave 2 wave 3 S-consistent I-consistent A-consistent characterized as consistently selfish. A detailed discussion of individual (within-game) consistency and the dynamics of behavior can be found in S2 File.

Observation S1-3
One third of the subjects behave consistently across games and over all three waves; only one of them reveals consistent other-regarding behavior in the last wave.