Otro M´etodo de Mec´anica Cu´antica: Teor´ıa DFT

The previous section discussed the mean values of actual waiting times. However, the relevance of the equilibrium selection result depends on the probability that a shift between equilibria is observed within a reasonable time scale. If the distribution of waiting times were highly skewed with

a relatively large mass attached to the lower range of waiting times, the shift between equilibria may occur within a reasonably short time span even though the mean value may be large. One of the main advantages of using simulation is th at I am able to observe distributions. In this section, there­ fore, the focus is on the distributions of waiting times using both analytic results and the simulation results of the previous section. There are three parts to the results in this section: first, the distribution of waiting times for a fixed island size is compared to the distribution for a model with uniform matching. Second, the probability that the shift between equilibria occurs within a fixed time is calculated for the various models. Finally, the vari­ ances when islands are on a circle are compared to those when islands are randomly matched.

It is however, worth discussing what the variance represents in this model. An approximation to the death game is used to calculate waiting times, and this means that the variance will be underestimated because the variance th at would arise from the game actually being played is ignored.

In the case of Kandori et al, actual waiting times are approximated with high precision by geometric distributions when A is small. This approxima­ tion becomes exact at the limit A ^ 0. Therefore, from whatever state the system is first observed, the probability th at the system escapes from the basin of attraction of th at state within a fixed period is calculated only us­ ing the mutation rate and the length of time from the current period. This arises because the system shows a jump between equilibria only if there are a number of simultaneous mutations. This stationarity in the distribution implies th at the expected waiting time is not very informative as to how long the system will actually take for a movement between equilibria. Station­ arity implies th at the expected waiting time is independent of the history of the system. In Binmore et al and in the current paper, the state of the system contains information about the expected waiting time because the

the system can be in a large number of different states.

When there is uniform interaction, interaction is with a wider pool of players and this makes the distribution of waiting times flatter. This is shown in the barcharts in figure 5.4. The main point made by figure 5.4, however, is that the distribution of waiting times in both cases is highly skewed.

F ig u re 5.4; S im u lated D istrib u tio n s o f W aitin g T im es

c 3 0 . 1 5 a- (a) W aiting tim es (b) W aiting tim es

Figure (a) shows the simulated distribution when there is only one island, figure (b) when island size is 10 and 7 = 0.15. In both figures, N = 100. The intervals on each chart are 15% of the mean of that distribution.

Figure 5.4 shows that the expected waiting time gives only limited in­ formation about when transitions will actually be observed. Therefore, to give some idea of when transitions will actually occur table 5.4 reports the empirical probability that the shift occurs within a fixed time for the various models.

Finally, I compare coefficients of variation from the random island model with the coefficients of variation from the circle model. Higher moments are not reported here because the difference in variances is more important.

The most important point made in table 5.5 is that the coefincients of variation of the circle model are consistently lower than the coefficients of variation of the random island model. This result was mentioned in section 5 . 3 above: it arises because of the more steady, but slower dispersion in the circle model due to the insulation provided by the circle structure. Further, the coefficients of variation are greater when the island size is larger. This

T able 5.4; P ro b a b ility o f a Shift to R isk -D o m in a n ce Time Model 1 0 0 0 2 0 0 0 3000 4000 Island Size 10 (595) 0.77 0.95 0.99 1 . 0 Island Size 20 (988) 0.58 0.82 0.93 0.970 Binmore et al (5029) 0.18 0.32 0.42 0.54 Kandori et a i 0 . 0 0 0 . 0 0 0 . 0 0 0 . 0 0

In all cases, N = 100, and for the island model, 7 = 0.15. Waiting times are given in brackets. Probabilities are simulated for the values for the island model, but the other probabilities are calculated analytically.

Table 5.5: Coefficients o f Variation o f W aiting Tim es

Number of Players Model 1 0 0 2 0 0 400 Random: size 10 0.86 (691) 0.72 (566) 0.59 (473) Circle: size 10 0.70 (551) 0.46 (532) 0.33 (643) Random: size 20 0.97 (1187) 0.90 (1342) 0.89 (1644) Circle: size 20 0.95 (1089) 0.80 (700) 0.68(662)

is because the difficulty of escaping the basin of attraction increases with island size and this increases the variance in times for the islands to convert.