We first conducted systematic computer simulations starting with an initial population of players with randomly generated strategies using a preliminary version of the model that does not allow repetition of games (see Methods). Parameter settings we used were as follows: population size n = 1000; mutation range ; length of a simulation run = 10,000 generations.

Figure 1 summarizes the simulation results for varying σ. Compared to the original UG (σ = 0), the characteristic offer amount of proposers ( ) increased greatly for higher values of σ. This is because the probabilistic fluctuations of responders' decisions increase the risk of rejection and the proposers will thus need to increase their offer amounts in order to avoid being rejected. In other words, fairness in offer amounts evolved just by increasing randomness in players' decision making. In this experiment, the value of σ were assumed to be the same for both proposers and responders. We also studied cases where σ is different between proposers and responders; see Supplementary Information (Fig. S3) for more details. However, this modification did not alter the qualitative interpretation of the results compared to cases with an identical σ value. Therefore, we used the same σ value for both proposers and responders hereafter.

Figure 1 Simulation results obtained using the preliminary model. Averages of characteristic offer amounts ( ) and acceptance thresholds ( ) are plotted over varying σ. Each data point is obtained by averaging p and q among all the players over the last 2,000 generations of each simulation run and then averaging the measurement over 10 independent simulation runs. Error bars represent standard deviations. The average payoffs and the typical simulation runs are shown in Supplementary Information (Figs. S1 and S2, respectively). Full size image

In the meantime, Figure 1 also shows that increasing σ does not affect evolution of the characteristic acceptance threshold ( ). To explore possibilities for both and to evolve toward higher values, we introduced an additional assumption to the preliminary model. Specifically, we introduced a new parameter, r, a probability for players to repeat playing the game again if an offer is rejected. We call this full version the Not Quite Ultimatum Game (NQUG) model (see Methods for details). Note that the repetition of games makes sense only in the UG with probabilistic decision making, but not in the original UG, because repeating games do not produce any different outcome in the original UG where the players' decisions are made deterministically.

It has been reported that fairness is likely to appear in repeated UG based on reciprocity17. Compared to that, our NQUG model is simply based on probabilistic decision making and does not require any information about the past that would be needed for reciprocity to function.

Figure 2 shows simulation results with the NQUG model, illustrating how the new parameter r influences the evolution of characteristic offer amounts ( , Fig. 2A) and characteristic acceptance thresholds ( , Fig. 2B). All the other parameters were set to the same values as used in Fig. 1. It was observed that, when r is relatively high (i.e., r ~ 0.8 or higher), the characteristic acceptance thresholds evolved toward a higher value when σ is positive (Fig. 2B), because rejection may become a more attractive alternative than accepting low offers for responders if repeating the game is possible. Consequently, proposers' characteristic offer amounts also evolved toward a comparably higher value (Fig. 2A), although this trend does not continue to hold for higher σ, where responders tend to accept low offers frequently even though their thresholds are high. We also note that, when r is moderate (i.e. 0.10 ≤ r ≤ 0.80), the general trend of is slightly decreasing (Fig. 2A and Fig. S6). This can be understood as follows. For r = 0, the proposers need to increase the offer amount in order to avoid rejection, as described in the preliminary model (Fig. 1). However, when r is moderately positive, there is a reasonable chance that a game is repeated again even if the offer is rejected. This makes it possible for the proposers to act more boldly and thus reduces the offer amount a little, because even low offers may sometimes be accepted if the game can be repeated. Nevertheless, when r is very high, it works for responders' benefit, as described above. In such cases, proposers have to increase their offer amount again to adapt to the responders' high demands. This is why a non-monotonic behavior is observed for when r is varied from 0 to 1 (Fig. S6).

Figure 2 Simulation results obtained using the NQUG model. Averages of characteristic offer amounts ( , A) and acceptance thresholds ( , B) are plotted for varying r and σ. As in Fig. 1, each data point is obtained by averaging p and q among all the players over the last 2,000 generations of each simulation run and then averaging the measurement over 10 independent simulation runs. Error bars represent standard deviations. The average payoffs and the typical simulation runs are shown in Supplementary Information (Figs. S4 and S5, respectively). Both and evolved to high values when r is high. Additional results of a more comprehensive parameter sweep experiment are provided in Supplementary Information (Fig. S6). Full size image

Figure 3 summarizes the results presented in Figs. 2A and 2B in a single - space, showing how the evolved strategy changes when σ and r are varied. When σ = 0 (Fig. 3, bottom-left), the game is equivalent to the original UG because repetition of game play does not make any difference in this case. Therefore, the players' average strategy always converges to the most rational behavior (low , nearly zero ) regardless of r. As σ increases, however, the average strategy shifts rightward toward higher . For r ~ 0.7 or less, the increase of is not significant, but for r ~ 0.8 and higher, the average strategy moves diagonally along the “perfect empathy” line ( = ), which indicates the evolution of fairness. It is also observed for higher σ and extremely high r (r ~ 1) that players tend to evolve to become “naysayers” (Fig. 3, top-right), always asking for more than what they would offer if they were proposers ( < ) because the rejection probability is nearly zero.