Curves for CJT not passing through (0.5, 0.5)

[rmaniv]

For some n values, the majority-voter curves don't pass through (0.5, 0.5) like they do on the plot published on CJT's Wikipedia page.

For example, look at n = 12 in the three curves below:

Figure 1: Monte Carlo

Figure 2: Deterministic

Figure 3: Wikipedia

For ease, I'm attaching the relevant functions.

• montecarlo(p, n, m) and deterministic(p, n, m) from theorems/condorcet.py

def montecarlo(p, n, m): # m is 0 for simple majority
    trials = int(1e6)
    r = majority(n, m)
    if r is None:
        return None
    return np.sum(np.sum((np.random.random(size=(trials, n)) < p).astype(int), axis=1) >= r) / trials

def deterministic(p, n, m): # m is 0 for simple majority
    r = majority(n, m)
    if r == None: return None
    P = 0
    while r <= n:
        P += (comb(n, r) * (p**r) * ((1 - p)**(n-r)))
        r += 1
    return P

• majority(n, m=0) from utils.py that montecarlo and deterministic use

def majority(n, m=0):
    if m == 0:
        if n % 2:
            r = (n // 2) + 1
        else:
            r = n // 2
    elif m > 50 and m <= 100: r = ceil((m / 100) * n)
    else:
        print("correctness_simulation(p, n, m) where m = 0 for simple-majority but > 50 and <= 100 for super-majority or unanimity")
        r == None
    return r

• R code published on Wikipedia:

  # cf. e.g. 
  # Odd case: DOI:10.1016/0176-2680(93)90010-R
  # Even case with coinflip ties: DOI:10.1007/BF01718092
  library(tidyverse); options(scipen=999); library(ggthemes); library(extrafont)
  theme_set(theme_minimal()); theme_update(legend.position = "bottom", text=element_text(family="Lato"))
  
  f <- function(n,p) {
  h=floor(n/2)+1
  if(n %% 2) {
   return(sum(choose(n,seq(h,n)) * p^seq(h,n) * (1-p)^(n-seq(h,n))))
  } else {
   return((sum(choose(n,seq(h,n)) * p^seq(h,n) * (1-p)^(n-seq(h,n)))) +
     (choose(n,n/2) * p^(n/2) * (1-p)^(n/2)) * 1/2) 
  }
  }
  df = tibble(p = seq(0,1,0.01)) %>% rowwise() %>% 
  mutate("1" = f(1,p), "3" = f(3,p), "12" = f(12,p), "100" = f(100,p)) %>% 
  pivot_longer(c("1","3","12","100"), names_to="N", values_to="pg")
  
  df %>% ggplot(aes(p, pg, color=N)) + geom_line() +
  labs(title = element_blank(), x = element_blank(), y = element_blank()) + 
  scale_color_colorblind(breaks=c(1,3,12,100))
  
  ggsave("cjt.svg", width=1000, height=1000, units="px")

To me, the functions seem identical but I might be missing something.

[cmdCrusaderr]

hey , could you brief me more about the issue?

[rmaniv]

I had missed the 1/2 at the end of the even n value condition in the R code.