weighting_function_analysis.R

library(ggplot2)
library(dplyr)
library(scales)

# --- Parameters ---
Tval <- 30000
Cval <- 100000
k <- 0.5

# --- Sqrt-affine reward (continuous at T, finite slope)
w_sqrtaffine <- function(s, T, C, k) {
  s   <- pmax(0, s)
  s_c <- pmin(s, C)                     # clamp at cap C
  
  out <- sqrt(pmin(s_c, T))             # branch 1: 0..T
  idx <- s_c > T                        # branch 2: T..C
  out[idx] <- sqrt(T + k^2 * (s_c[idx] - T))
  
  out                                    # branch 3: s > C stays flat due to clamp
}

# --- Derivative (piecewise)
# Note: derivative undefined at s=0 (infinite); we return NA there for plotting.
dw_sqrtaffine <- function(s, T, C, k) {
  s <- pmax(0, s)
  
  out <- rep(NA_real_, length(s))
  
  # 1) 0 < s < T : d/ds sqrt(s) = 1/(2*sqrt(s))
  idx1 <- s > 0 & s < T
  out[idx1] <- 1 / (2 * sqrt(s[idx1]))
  
  # 2) T < s < C : d/ds sqrt(T + k^2 (s - T)) = k^2 / (2 * sqrt(T + k^2 (s - T)))
  idx2 <- s > T & s < C
  out[idx2] <- k^2 / (2 * sqrt(T + k^2 * (s[idx2] - T)))
  
  # 3) s > C : flat (cap), derivative 0
  idx3 <- s > C
  out[idx3] <- 0
  
  # At exactly s = T and s = C there are kinks (left/right derivatives differ)
  # We'll leave them as NA to show the kink in the plot.
  out[s == T | s == C] <- NA_real_
  
  out
}


# --- Grid and evaluations
s_grid <- seq(0, 1.1 * C, length.out = 2000)
w_val  <- w_sqrtaffine(s_grid, T, C, k)
dw_val <- dw_sqrtaffine(s_grid, T, C, k)

# --- Plots
df <- data.frame(s = s_grid, w = w_val, dw = dw_val)

p_levels <- ggplot(df, aes(s, w)) +
  geom_line(linewidth = 1) +
  geom_vline(xintercept = T, linetype = "dashed") +
  geom_vline(xintercept = C, linetype = "dashed") +
  labs(title = "Levels w(s)",
       x = "Self-stake s",
       y = "w(s)") +
  theme_minimal()

p_deriv <- ggplot(df, aes(s, dw)) +
  geom_line(linewidth = 1) +
  geom_vline(xintercept = T, linetype = "dashed") +
  geom_vline(xintercept = C, linetype = "dashed") +
  labs(title = "Marginal Reward dw/ds",
       x = "Self-stake s",
       y = "dw/ds") +
  theme_minimal()

p_levels
p_deriv