finalplots.Rmd

---
title: "Plots"
output:
  pdf_document:
    keep_tex: yes
  html_document: default
  html_notebook: default
---

```{r setup, include = F}
library(tikzDevice)
library(magrittr)
library(knitr)
library(data.table)
library(dplyr)
# library(readr)
# library(mvtnorm)
library(VGAM) # for erfc
library(rootSolve)
# library(plotly)
library(parallel) # for mcapply

opts_chunk$set(dev = 'tikz', 
               fig.path = './Overleaf/figures/', 
               echo = F, error = F, cache = F, message = F, warning = F,
               eval = F,
               # results='hide', 
               autodep = T,
               fig.height = 6)

options(stringsAsFactors = F
        # ,
        # tikzMetricPackages = c(
        #   "\\usepackage{amssymb, amsmath, mathrsfs, amsfonts, mathtools, amsthm}"
        #   )
        )

par(mar = c(0.5,0.5,0,0), oma = c(1,1,1,0), mgp = c(0.5, 0, 0), cex.axis = .9, cex = 2)

ncore = detectCores()
```

```{r functions, include = F, eval = T}
source('functions.R')

source('DefaultValues.R')
```

```{r util3_A_plot, include = T, eval = F}
for(Pstar in Pstar_vals){
  par(mar = c(2.6,2.6,0,0), oma = c(1,1,0,0), mgp = c(1.2, 0.2, 0), 
      cex.axis=1, cex = 3, tck = -0.02, 
      bty = 'l',
      font.main = 1)

  P3_lower_v = P3_lower(Pstar, alphas[1])
  
  pt3vs1 = seq(axis_lim[1],P3_lower_v,length.out = 200)
  pt3vs2 = seq(P3_lower_v, axis_lim[2],length.out = 200)
  
  pt3vs = c(pt3vs1, pt3vs2)
  
  plot(pt3vs, util3_A_cont(pt3vs), type = 'l',
       xlim = axis_lim,
       ylim = axis_lim,
       xlab = 'Token$_b$ price at $t_3$, $P_{t_3}$', ylab = "Alice's utility at $t_3$, $U_{t_3}^\\mathcal{A}$", 
       col = col_cont)
  
  title(main = paste('$P_* =', Pstar, '$'), line = -0.7, cex = 1)
  
  util3_A_stop_v = util3_A_stop(Pstar)
  
  lines(pt3vs, rep(util3_A_stop_v, length(pt3vs)))
  
  points(P3_lower_v, util3_A_stop_v, pch = '.', col = 'red', cex = 5)

}

legend('right', 
       pch = '.',
       bty = 'n',
       pt.cex = 5,
       col = 'red', 
       legend = '$\\underline{P_{t_3}}$'
       )

legend('bottomright', lty = 1, 
       seg.len = 0.5,
       bty = 'n',
       col = c(col_cont, 'black'), 
       legend = c(
         '$U_{t_3}^\\mathcal{A}(\\mathit{cont})$', 
         '$U_{t_3}^\\mathcal{A}(\\mathit{stop})$'
         )
       )
```

```{r util2_B_plot, include = T, eval = F}
for(Pstar in Pstar_vals){
  par(mar = c(2.6,2.6,0,0), oma = c(1,1,0,0), mgp = c(1.2, 0.2, 0), 
      cex.axis=1, cex = 3, tck = -0.02, 
      bty = 'l',
      font.main = 1)
  
  pt2vs = seq(1e-10,axis_lim[2],length.out = 200)
  
  util2_B_stop_v = util2_B_stop(pt2vs)
  util2_B_cont_v = sapply(pt2vs, function(x) util2_B_cont(x, Pstar))
  
  pt2threshsv = which(diff(util2_B_stop_v > util2_B_cont_v) !=0) # sign switch in the graph
  
  plot(pt2vs, util2_B_stop_v, type = 'l',
       xlim = axis_lim,
       ylim = axis_lim, 
       xlab = 'Token$_b$ price at $t_2$, $P_{t_2}$', ylab = "Bob's utility at $t_2$, $U_{t_2}^\\mathcal{B}$")
  lines(pt2vs, sapply(pt2vs, function(x) util2_B_cont(x, Pstar)), col = col_cont)
  
  points(
  c(mean(pt2vs[pt2threshsv[1]+(0:1)]), mean(pt2vs[pt2threshsv[2]+(0:1)])),
  c(mean(util2_B_stop_v[pt2threshsv[1]+(0:1)]), mean(util2_B_stop_v[pt2threshsv[2]+(0:1)])),
  col = c('red','blue'), pch = '.', cex = 5
  )
  
  
  title(main = paste('$P_* =', Pstar, '$'), line = -0.75, cex = 1)
}

  legend('right', 
         pch = '.',
         pt.cex = 5,
         col = c('blue','red'), 
         bty = 'n',
         legend = c('$\\overline{P_{t_2}}$', '$\\underline{P_{t_2}}$')
         )
  
  legend('bottomright', lty = 1,
         col = c(col_cont, 'black'),
         bty = 'n',
         seg.len = 0.5,
         legend = c(
           '$U_{t_2}^\\mathcal{B}(\\mathit{cont})$',
           '$U_{t_2}^\\mathcal{B}(\\mathit{stop})$'))
```

```{r util1_A_plot, include = T, eval = F}

# util1_A_cont_v = Vectorize(util1_A_cont)(Pstars)

# save(util1_A_cont_v, file = 'data/util1_A_cont_v.rda')
load('data/util1_A_cont_v.rda')

util1_A_stop_v = util1_A_stop(Pstars)

  par(mar = c(2.6,2.6,0,0), oma = c(1,1,0,0), mgp = c(1.2, 0.2, 0), 
      cex.axis=1, cex = 3, tck = -0.02, 
      bty = 'l',
      font.main = 1)

plot(Pstars, util1_A_stop_v, type = 'l',
     xlim = axis_lim,
     ylim = axis_lim,
     xlab = 'Exchange rate $P_*$',
     ylab = "Alice's utility at $t_1$, $U_{t_1}^\\mathcal{A}$")
lines(Pstars, util1_A_cont_v, col = col_cont)

p0boundsind = which(diff(Pstars < util1_A_cont_v)!=0)

points(
  c(mean(Pstars[p0boundsind[1]+(0:1)]), mean(Pstars[p0boundsind[2]+(0:1)])),
  c(mean(util1_A_stop_v[p0boundsind[1]+(0:1)]), mean(util1_A_stop_v[p0boundsind[2]+(0:1)])),
  col = c('red','blue'), pch = '.', cex = 5
  )

legend('right', 
         pch = '.',
         pt.cex = 5,
         col = c('blue','red'), 
         bty = 'n',
         legend = c('$\\overline{P_*}$', '$\\underline{P_*}$')
         )

legend('bottomright', lty = 1, 
       seg.len = 0.5,
       bty = 'n',
       col = c(col_cont, 'black'), 
       legend = c(
         '$U_{t_1}^\\mathcal{A}(\\mathit{cont})$', 
         '$U_{t_1}^\\mathcal{A}(\\mathit{stop})$'
         )
       )
```

```{r util1_A_unknx_plot, include = T, eval = T}
# util1_A_unknx_v = Vectorize(util1_A_unknx)(Pstars)
# 
# save(util1_A_unknx_v, file = 'data/util1_A_unknx_v.rda')

load('data/util1_A_unknx_v.rda')

Pstars_test = c(0.01, 1, 2, 5, 10, 20, 25, 100)
util1_A_unknx_test = c(
  -0.001306397,
  0.06348887,
  0.126816,
  0.3143151,
  0.4173118,
  -1.650157,
  -2.785154,
  -13.06417
)

Pstars_add = c(seq(0.011, 0.19, length.out = 5),
               seq(3.3, 19, length.out = 15))

# util1_A_unknx_v_add = Vectorize(util1_A_unknx)(Pstars_add)
#
# save(util1_A_unknx_v_add, file = 'data/util1_A_unknx_v_add.rda')

load('data/util1_A_unknx_v_add.rda')

odr = c(Pstars, Pstars_test, Pstars_add) %>% order
ps = c(Pstars, Pstars_test, Pstars_add)[odr]
util1_A_unknx_vs = c(util1_A_unknx_v, util1_A_unknx_test, util1_A_unknx_v_add)[odr]

nullpoint = approx(x = util1_A_unknx_vs, y = ps, xout = 0)$y
hightpoint = ps[which.max(util1_A_unknx_vs)]
par(
  mar = c(2.6, 2.6, 0, 0),
  oma = c(1, 1, 0, 0),
  mgp = c(1.1, 0.2, 0),
  cex.axis = 1,
  cex = 2.7,
  tck = -0.02,
  bty = 'l',
  font.main = 1
)


plot(
  ps,
  util1_A_unknx_vs,
  type = 'l',
  xlim = c(1e-10,13),
  ylim = c(-0.05,0.45),
  xlab = 'Amount Token$_a$ locked, $P_*$',
  ylab = "Alice's utility at $t_1$, $U_{t_1, \\mathrm{x}}^\\mathcal{A}$"
)
abline(h = 0, lty = 2, col = 'grey')

points(
  c(nullpoint, hightpoint),
  c(0, max(util1_A_unknx_vs)),
  col = c('red','blue'), pch = '.', cex = 5
  )

legend('topleft', 
         pch = '.',
         pt.cex = 5,
         col = c('blue','red'), 
         bty = 'n',
         legend = c('$\\overline{P_*}$', '$\\underline{P_*}$')
         )

```

```{r unknx_X_opt_plot, include = T, eval = T}
pt2vs = seq(1e-10, 13, length.out = 600)

par(
  mar = c(2.6, 2.6, 0, 0),
  oma = c(1, 1, 0, 0),
  mgp = c(1.2, 0.2, 0),
  cex.axis = 1,
  cex = 2.7,
  tck = -0.02,
  bty = 'l',
  font.main = 1
)

Pstar_vs = c(nullpoint %>% round(2), 4, hightpoint %>% round(2))
# X_opt_v = Vectorize(X_opt)(pt2vs, Pstar = Pstar_vs[1])
# X_opt_v_2 = Vectorize(X_opt)(pt2vs, Pstar = Pstar_vs[2])
# X_opt_v_3 = Vectorize(X_opt)(pt2vs, Pstar = Pstar_vs[3])
# 
# save(X_opt_v, X_opt_v_2, X_opt_v_3, file = 'data/X_opt_v.Rda')
load('data/X_opt_v.Rda')
plot(
  pt2vs,
  X_opt_v,
  type = 'l',
  xlim = c(1e-10,13),
  ylim = c(0, 5),
  xlab = 'Token$_b$ price at $t_2$, $P_{t_2}$',
  ylab = "Optimal Token$_b$ $X_*$",
  col = 'red',
  lwd = lwd_success
)

lines(pt2vs,
      X_opt_v_2,
      col = col_success[1],
      lty = 2,
      lwd = lwd_success)

lines(pt2vs,
      X_opt_v_3,
      col = 'blue',
      lty = 3,
      lwd = lwd_success)


legend(
  'topright',
  legend = Pstar_vs,
  col = c('red', col_success[1], 'blue'),
  lty = c(1, 3, 2),
  lwd = lwd_success,
  bty = 'n',
  title = 'Token$_a$ amount $P_*$'
)
```

```{r success_val_unknx, include = T, eval = T}
Pstars_unknx_no = 40
Pstars_unknx = seq(0.01, 17, length.out = Pstars_unknx_no)
# sucrate_v_unknx = Vectorize(sucrate_unknx)(Pstars_unknx)
# save(sucrate_v_unknx, file = 'data/sucrate_v_unknx.rda')
load('data/sucrate_v_unknx.rda')

sucrate_unknx_vs = (1:Pstars_unknx_no) %>% sapply(function(x)
  sucrate_v_unknx[, x]$value)

sucrate_v_unknx_add = list()
Pstars_unknx_add = c(0.015, 0.11125, 0.01200, 0.05650)
# for(i in 3:length(Pstars_unknx_add)){
#   print(i)
#   sucrate_v_unknx_add[[i]] = sucrate_unknx(Pstars_unknx_add[i])
#   print(sucrate_v_unknx_add[[i]])
# }
# save(sucrate_v_unknx_add, file = 'data/sucrate_v_unknx_add.rda')
load('data/sucrate_v_unknx_add.rda')
sucrate_unknx_vs_add = (1:length(Pstars_unknx_add)) %>% sapply(function(x)
  sucrate_v_unknx_add[[x]]$value)

ps = c(Pstars_unknx, Pstars_unknx_add)
odr = ps %>% order
ps = ps[odr]
suc_unknx_vs = c(sucrate_unknx_vs, sucrate_unknx_vs_add)[odr]

# only feasible points
sel = ps > nullpoint & ps < hightpoint

load('data/sucrate_v.rda')

par(
  mar = c(2.6, 2.6, 0, 0),
  oma = c(1, 1, 0, 0),
  mgp = c(1.1, 0.2, 0),
  cex.axis = 1,
  cex = 2.7,
  tck = -0.02,
  bty = 'l',
  font.main = 1
)

# plot(
#   Pstars,
#   sucrate_v,
#   type = 'l',
#   xlab = 'Exchange rate $P_*$',
#   ylab = "Success rate $\\mathit{SR}$",
#   xlim = axis_lim,
#   ylim = c(0, 1.05),
#   col = col_success[2],
#   lwd = lwd_success
# )

plot(
  Pstars,
  sucrate_v,
  type = 'l',
  xlim = c(1e-10, 8.9),
  ylim = c(0, 1.05),
  col = col_success[2],
  xlab = 'Amount Token$_a$ locked $P_*$',
  ylab = "Success rate $\\mathit{SR}$",
  lwd = lwd_success
)

lines(
  ps[sel],
  suc_unknx_vs[sel],
  col = col_success[1],
  lty = 2,
  lwd = lwd_success
)

legend(
  'bottomleft',
  legend = c('basic setup', 'uncertain exchange'),
  col = col_success[c(2, 1)],
  lty = c(1, 2),
  lwd = lwd_success,
  bty = 'n',
  # title = 'Collateral $Q$'
)
```

```{r success_val, include = T, eval = F}
# sucrate_v = mclapply(Pstars, sucrate) %>% unlist
# save(sucrate_v, file = 'data/sucrate_v.rda')
load('data/sucrate_v.rda')
```

```{r success_plot_alphaA, include = T, eval = F}
# reset everything
source('DefaultValues.R')

alphaAs = c(0.1,0.2, alphas[1],0.4)

# alphas[1] = alphaAs[2]
# sucrate_v_alphaA_1 = mclapply(Pstars, sucrate) %>% unlist
# 
# alphas[1] = alphaAs[1]
# sucrate_v_alphaA_2 = mclapply(Pstars, sucrate) %>% unlist
# 
# alphas[1] = alphaAs[4]
# sucrate_v_alphaA_3 = mclapply(Pstars, sucrate) %>% unlist
# 
# save(sucrate_v_alphaA_1, sucrate_v_alphaA_2, sucrate_v_alphaA_3, 
#      file = 'data/sucrate_v_alphaA.rda')
load('data/sucrate_v_alphaA.rda')

par(mar = c(2.6,2.6,0,0), oma = c(1,1,0,0), mgp = c(1.1, 0.2, 0), 
    cex.axis=1, cex = 2.7, tck = -0.02, 
    bty = 'l',
    font.main = 1)

plot(Pstars,sucrate_v,type = 'l', 
     xlab = 'Exchange rate $P_*$',
     ylab = "Success rate $\\mathit{SR}$",
     xlim = axis_lim,
     ylim = c(0,1.05),
     col = col_success[2], lwd = lwd_success)
lines(Pstars, sucrate_v_alphaA_1, col = col_success[1], lty = 2, lwd = lwd_success)
# lines(Pstars, sucrate_v_alphaA_2, col = 'red') #only NAs
lines(Pstars, sucrate_v_alphaA_3, col = col_success[3], lty = 3, lwd = lwd_success)

legend('topleft',
       legend = alphaAs,
       col = c('grey', col_success),
       lty = c(NA, 2,1,3),
       pch = c(0, NA, NA, NA),
       lwd = lwd_success,
       bty = 'n',
       title = '$\\alpha^{\\mathcal{A}}$'
)
```

```{r success_plot_alphaB, include = T, eval = F}
# reset everything
source('DefaultValues.R')

alphaBs = c(0.1,0.2, alphas[2],0.4)

# alphas[2] = alphaBs[2]
# sucrate_v_alphaB_1 = mclapply(Pstars, sucrate) %>% unlist
# 
# alphas[2] = alphaBs[1]
# sucrate_v_alphaB_2 = mclapply(Pstars, sucrate) %>% unlist
# 
# alphas[2] = alphaBs[4]
# sucrate_v_alphaB_3 = mclapply(Pstars, sucrate) %>% unlist
# 
# save(sucrate_v_alphaB_1, sucrate_v_alphaB_2, sucrate_v_alphaB_3,
#      file = 'data/sucrate_v_alphaB.rda')
load('data/sucrate_v_alphaB.rda')

par(mar = c(2.6,2.6,0,0), oma = c(1,1,0,0), mgp = c(1.1, 0.2, 0), 
    cex.axis=1, cex = 2.7, tck = -0.02, 
    bty = 'l',
    font.main = 1)

plot(Pstars,sucrate_v,type = 'l', 
     xlab = 'Exchange rate $P_*$',
     ylab = "Success rate $\\mathit{SR}$",
     xlim = axis_lim,
     ylim = c(0,1.05),
     col = col_success[2], lwd = lwd_success)
lines(Pstars, sucrate_v_alphaB_1, col = col_success[1], lty = 2, lwd = lwd_success)
# lines(Pstars, sucrate_v_alphaA_2, col = 'red') #only NAs
lines(Pstars, sucrate_v_alphaB_3, col = col_success[3], lty = 3, lwd = lwd_success)

legend('topleft',
       legend = alphaBs,
       col = c('grey', col_success),
       lty = c(NA, 2,1,3),
       pch = c(0, NA, NA, NA),
       lwd = lwd_success,
       bty = 'n',
       title = '$\\alpha^{\\mathcal{B}}$'
)
```

```{r success_plot_rA, include = T, eval = F}
# reset everything
source('DefaultValues.R')

rAs = c(0.005, rs[1], 0.015, 0.02)

# rs[1] = rAs[1]
# sucrate_v_rA_1 = mclapply(Pstars, sucrate, mc.cores = ncore) %>% unlist
# 
# rs[1] = rAs[3]
# sucrate_v_rA_2 = mclapply(Pstars, sucrate, mc.cores = ncore) %>% unlist
# 
# rs[1] = rAs[4]
# sucrate_v_rA_3 = mclapply(Pstars, sucrate, mc.cores = ncore) %>% unlist
# 
# save(sucrate_v_rA_1, sucrate_v_rA_2, sucrate_v_rA_3,
#      file = 'data/sucrate_v_rA.rda')
load('data/sucrate_v_rA.rda')

par(mar = c(2.6,2.6,0,0), oma = c(1,1,0,0), mgp = c(1.1, 0.2, 0), 
    cex.axis=1, cex = 2.7, tck = -0.02, 
    bty = 'l',
    font.main = 1)

plot(Pstars,sucrate_v,type = 'l', 
     xlab = 'Exchange rate $P_*$',
     ylab = "Success rate $\\mathit{SR}$",
     xlim = axis_lim,
     ylim = c(0,1.05),
     col = col_success[2], lwd = lwd_success)
lines(Pstars, sucrate_v_rA_1, col = col_success[1], lty = 2, lwd = lwd_success)
lines(Pstars, sucrate_v_rA_2, col = col_success[3], lty = 3, lwd = lwd_success)

legend('topleft',
       legend = rAs,
       col = c(col_success, 'grey'),
       lty = c(2,1,3, NA),
       pch = c(NA, NA, NA, 0),
       lwd = lwd_success,
       bty = 'n',
       title = '$r^{\\mathcal{A}}$'
)
```

```{r success_plot_rB, include = T, eval = F}
# reset everything
source('DefaultValues.R')

rBs = c(0.005, rs[2], 0.02, 0.03)

# rs[2] = rBs[1]
# sucrate_v_rB_1 = mclapply(Pstars, sucrate, mc.cores = ncore) %>% unlist
# 
# rs[2] = rBs[3]
# sucrate_v_rB_2 = sucrate_v_rB_3 mclapply(Pstars, sucrate, mc.cores = ncore) %>% unlist
# 
# rs[2] = rBs[4]
# sucrate_v_rB_3 = mclapply(Pstars, sucrate, mc.cores = ncore) %>% unlist

# save(sucrate_v_rB_1, sucrate_v_rB_2, sucrate_v_rB_3,
#      file = 'data/sucrate_v_rB.rda')
load('data/sucrate_v_rB.rda')

par(mar = c(2.6,2.6,0,0), oma = c(1,1,0,0), mgp = c(1.1, 0.2, 0), 
    cex.axis=1, cex = 2.7, tck = -0.02, 
    bty = 'l',
    font.main = 1)

plot(Pstars,sucrate_v,type = 'l', 
     xlab = 'Exchange rate $P_*$',
     ylab = "Success rate $\\mathit{SR}$",
     xlim = axis_lim,
     ylim = c(0,1.05),
     col = col_success[2], lwd = lwd_success)
lines(Pstars, sucrate_v_rB_1, col = col_success[1], lty = 2, lwd = lwd_success)
lines(Pstars, sucrate_v_rB_2, col = col_success[3], lty = 3, lwd = lwd_success)

legend('topleft',
       legend = rBs,
       col = c(col_success, 'grey'),
       lty = c(2,1,3, NA),
       pch = c(NA, NA, NA, 0),
       lwd = lwd_success,
       bty = 'n',
       title = '$r^{\\mathcal{B}}$'
)
```

```{r success_plot_taua, include = T, eval = F}
# reset everything
source('DefaultValues.R')

tauas = c(1, taua, 5, 7)
# 
# taua = tauas[1]
# sucrate_v_taua_1 = mclapply(Pstars, sucrate, mc.cores = ncore) %>% unlist
# 
# taua = tauas[3]
# sucrate_v_taua_2 = mclapply(Pstars, sucrate, mc.cores = ncore) %>% unlist
# 
# taua = tauas[4]
# sucrate_v_taua_3 = mclapply(Pstars, sucrate, mc.cores = ncore) %>% unlist
# 
# save(sucrate_v_taua_1, sucrate_v_taua_2, sucrate_v_taua_3,
#      file = 'data/sucrate_v_taua.rda')
load('data/sucrate_v_taua.rda')

par(mar = c(2.6,2.6,0,0), oma = c(1,1,0,0), mgp = c(1.1, 0.2, 0), 
    cex.axis=1, cex = 2.7, tck = -0.02, 
    bty = 'l',
    font.main = 1)

plot(Pstars,sucrate_v,type = 'l', 
     xlab = 'Exchange rate $P_*$',
     ylab = "Success rate $\\mathit{SR}$",
     xlim = axis_lim,
     ylim = c(0,1.05),
     col = col_success[2], lwd = lwd_success)
lines(Pstars, sucrate_v_taua_1, col = col_success[1], lty = 2, lwd = lwd_success)
lines(Pstars, sucrate_v_taua_2, col = col_success[3], lty = 3, lwd = lwd_success)

legend('topleft',
       legend = tauas,
       col = c(col_success, 'grey'),
       lty = c(2,1,3, NA),
       pch = c(NA, NA, NA, 0),
       lwd = lwd_success,
       bty = 'n',
       title = '$\\tau_a$'
)
```

```{r success_plot_taub, include = T, eval = F}
# reset everything
source('DefaultValues.R')

taubs = c(2, taub, 6, 8)

# taub = taubs[1]
# sucrate_v_taub_1 = mclapply(Pstars, sucrate, mc.cores = ncore) %>% unlist
# 
# taub = taubs[3]
# sucrate_v_taub_2 = mclapply(Pstars, sucrate, mc.cores = ncore) %>% unlist
# 
# taub = taubs[4]
# sucrate_v_taub_3 = mclapply(Pstars, sucrate, mc.cores = ncore) %>% unlist

# save(sucrate_v_taub_1, sucrate_v_taub_2, sucrate_v_taub_3,
#      file = 'data/sucrate_v_taub.rda')
load('data/sucrate_v_taub.rda')

par(mar = c(2.6,2.6,0,0), oma = c(1,1,0,0), mgp = c(1.1, 0.2, 0), 
    cex.axis=1, cex = 2.7, tck = -0.02, 
    bty = 'l',
    font.main = 1)

plot(Pstars,sucrate_v,type = 'l', 
     xlab = 'Exchange rate $P_*$',
     ylab = "Success rate $\\mathit{SR}$",
     xlim = axis_lim,
     ylim = c(0,1.05),
     col = col_success[2], lwd = lwd_success)
lines(Pstars, sucrate_v_taub_1, col = col_success[1], lty = 2, lwd = lwd_success)
lines(Pstars, sucrate_v_taub_2, col = col_success[3], lty = 3, lwd = lwd_success)

legend('topleft',
       legend = taubs,
       col = c(col_success, 'grey'),
       lty = c(2,1,3, NA),
       pch = c(NA, NA, NA, 0),
       lwd = lwd_success,
       bty = 'n',
       title = '$\\tau_b$'
)
```

```{r success_plot_sigma, include = T, eval = F}
# reset everything
source('DefaultValues.R')

sigmas = c(0.05, sigma, 0.15, 0.2)

# sigma = sigmas[1]
# sucrate_v_sigma_1 = mclapply(Pstars, sucrate, mc.cores = ncore) %>% unlist

# sigma = sigmas[3]
# sucrate_v_sigma_2 = mclapply(Pstars, sucrate, mc.cores = ncore) %>% unlist
# 
# sigma = sigmas[4]
# sucrate_v_sigma_3 = mclapply(Pstars, sucrate, mc.cores = ncore) %>% unlist

# save(sucrate_v_sigma_1, sucrate_v_sigma_2, sucrate_v_sigma_3,
#      file = 'data/sucrate_v_sigma.rda')
load('data/sucrate_v_sigma.rda')

par(mar = c(2.6,2.6,0,0), oma = c(1,1,0,0), mgp = c(1.1, 0.2, 0), 
    cex.axis=1, cex = 2.7, tck = -0.02, 
    bty = 'l',
    font.main = 1)

plot(Pstars,sucrate_v,type = 'l', 
     xlab = 'Exchange rate $P_*$',
     ylab = "Success rate $\\mathit{SR}$",
     xlim = axis_lim,
     ylim = c(0,1.05),
     col = col_success[2], lwd = lwd_success)
lines(Pstars, sucrate_v_sigma_1, col = col_success[1], lty = 2, lwd = lwd_success)
lines(Pstars, sucrate_v_sigma_2, col = col_success[3], lty = 3, lwd = lwd_success)

legend('topleft',
       legend = sigmas,
       col = c(col_success, 'grey'),
       lty = c(2,1,3, NA),
       pch = c(NA, NA, NA, 0),
       lwd = lwd_success,
       bty = 'n',
       title = '$\\sigma$'
)
```

```{r success_plot_mu, include = T, eval = F}
# reset everything
source('DefaultValues.R')

mus = c(-0.002, 0, mu, 0.004)

# mu = mus[1]
# sucrate_v_mu_1 = mclapply(Pstars, sucrate, mc.cores = ncore) %>% unlist
# 
# mu = mus[2]
# sucrate_v_mu_2 = mclapply(Pstars, sucrate, mc.cores = ncore) %>% unlist
# 
# mu = mus[4]
# sucrate_v_mu_3 = mclapply(Pstars, sucrate, mc.cores = ncore) %>% unlist

# save(sucrate_v_mu_1, sucrate_v_mu_2, sucrate_v_mu_3,
#      file = 'data/sucrate_v_mu.rda')
load('data/sucrate_v_mu.rda')

par(mar = c(2.6,2.6,0,0), oma = c(1,1,0,0), mgp = c(1.1, 0.2, 0), 
    cex.axis=1, cex = 2.7, tck = -0.02, 
    bty = 'l',
    font.main = 1)

plot(Pstars,sucrate_v,type = 'l', 
     xlab = 'Exchange rate $P_*$',
     ylab = "Success rate $\\mathit{SR}$",
     xlim = axis_lim,
     ylim = c(0,1.05),
     col = col_success[2], lwd = lwd_success)
lines(Pstars, sucrate_v_mu_1, col = 'yellow', lty = 2, lwd = lwd_success)
lines(Pstars, sucrate_v_mu_2, col = col_success[1], lty = 3, lwd = lwd_success)
lines(Pstars, sucrate_v_mu_3, col = col_success[3], lty = 4, lwd = lwd_success)

legend('topleft',
       legend = mus,
       col = c('yellow', col_success),
       lty = c(4, 2,1,3),
       # pch = c(NA, NA, NA, 0),
       lwd = lwd_success,
       bty = 'n',
       title = '$\\mu$'
)
```

```{r alphalower, include = T, eval = F}

alphaAls = rep(NA, datanumber)

for (i in 4:datanumber){
  try(
    alphaAls[i] <- alphal(PstarAls[i])$root, silent = T
  )
}

save(alphaAls, file = "data/alphaAls.rda")

load('data/alphaAls.rda')

par(mar = c(2.6,2.6,0,0), oma = c(1,1,0,0), mgp = c(1.1, 0.2, 0), 
    cex.axis=1, cex = 2.7, tck = -0.02, 
    bty = 'l',
    font.main = 1)

alphaAls[is.na(alphaAls)] = 0.01 
plot(PstarAls, alphaAls, type = 'l', 
     xlim = axis_lim,
     lwd = 5,
     # pch = 4, cex =0.2, 
     xlab='Exchange rate $P_*$', 
     ylab = "$\\alpha^\\mathcal{A}$'s lower bound, $\\alpha_l^\\mathcal{A}$")
# abline(v = pt1, lty =2)
# text(x=pt1, y =0, 'Price at t1')

abline(h = alphas[1])
```

```{r success_unc, include = T, eval = F}
# reset everything
load('data/alphaAls.rda')
alphaAls[is.na(alphaAls)] = 0.01 
ind = which(alphas[1] > alphaAls)

sucrate_unc_foo = function(i){
  sucrate_unc(Pstar = PstarAls[i], alpha_lower = alphaAls[i])
}

# sucrate_v_unc_alpha = mclapply(ind, sucrate_unc_foo) %>% unlist
# 
# save(sucrate_v_unc_alpha, file = 'data/sucrate_v_unc_alpha.rda')
load('data/sucrate_v_unc_alpha.rda')

par(mar = c(2.6,2.6,0,0), oma = c(1,1,0,0), mgp = c(1.1, 0.2, 0), 
    cex.axis=1, cex = 2.7, tck = -0.02, 
    bty = 'l',
    font.main = 1)

plot(PstarAls[ind], sucrate_v_unc_alpha, col = col_success[3], type = 'l',
     lty = 2,
     xlab = 'Exchange rate $P_*$',
     ylab = "Success rate $\\mathit{SR}$",
     xlim = axis_lim,
     ylim = c(0,1.05), 
     lwd = lwd_success)
lines(Pstars[is.finite(sucrate_v)], sucrate_v[is.finite(sucrate_v)],
     col = col_success[2],
     lwd = lwd_success)

points(Pstars[is.finite(sucrate_v)][5], sucrate_v[is.finite(sucrate_v)][5],cex = 3)
# legend('topleft',
#        legend = alphaAs,
#        col = c('grey', col_success),
#        lty = c(NA, 2,1,3),
#        pch = c(0, NA, NA, NA),
#        lwd = lwd_success,
#        bty = 'n',
#        title = '$\\alpha^{\\mathcal{A}}$'
# )
```

```{r util2_B_plot_col, include = T, eval = F}
par(mar = c(2.6,2.6,0,0), oma = c(1,1,0,0), mgp = c(1.2, 0.2, 0), 
      cex.axis=1, cex = 3, tck = -0.02, 
      bty = 'l',
      font.main = 1)

axis_lim = c(1e-10,3.3)

for(Q in Qs){

for(Pstar in Pstar_vals){
  
  pt2vs = seq(1e-10,axis_lim[2],length.out = 200)
  
  util3_B_stop_v = util2_B_stop(pt2vs)
  util3_B_cont_v = sapply(pt2vs, function(x) util2_B_cont_col(x, Pstar, colla = Q))
  
  pt2threshsv = which(diff(util3_B_stop_v > util3_B_cont_v) !=0) # sign switch in the graph
  
  plot(pt2vs, util3_B_stop_v, type = 'l',
       xlim = axis_lim,
       ylim = axis_lim, 
       xlab = 'Token$_b$ price at $t_2$, $P_{t_2}$', ylab = "Bob's utility at $t_2$, $U_{t_2}^\\mathcal{B}$")
  lines(pt2vs, util3_B_cont_v, col = col_cont)
  
  points(sapply(pt2threshsv, function(x) mean(pt2vs[x+(0:1)])),
         sapply(pt2threshsv, function(x) mean(util3_B_stop_v[x+(0:1)])),
  col = c('black'), pch = '.', cex = 5
  )
  
  
  title(main = paste('$P_* =', Pstar, '$'), line = -0.75, cex = 1)
}

  
  legend('bottomright', lty = 1,
         col = c(col_cont, 'black'),
         bty = 'n',
         seg.len = 0.5,
         legend = c(
           '$U_{t_2,\\mathrm{c}}^\\mathcal{B}(\\mathit{cont})$',
           '$U_{t_2}^\\mathcal{B}(\\mathit{stop})$'))
  
}
```

```{r util1_plot_col, include = T, eval = F}
# util1_A_cont_v_col1 = Vectorize(function(x) util1_A_cont_col(x, pt1 = P1, alphaA = alphas[1], alphaB = alphas[2], colla = Qs[1]))(Pstars)
# util1_A_cont_v_col2 = Vectorize(function(x) util1_A_cont_col(x, pt1 = P1, alphaA = alphas[1], alphaB = alphas[2], colla = Qs[2]))(Pstars)
# 
# save(util1_A_cont_v_col1, util1_A_cont_v_col2, file = 'data/util1_A_cont_v_col.rda')
# 
# util1_B_cont_v_col1 = Vectorize(function(x) util1_B_cont_col(x, pt1 = P1, alphaB = alphas[2], colla = Qs[1]))(Pstars)
# util1_B_cont_v_col2 = Vectorize(function(x) util1_B_cont_col(x, pt1 = P1, alphaB = alphas[2], colla = Qs[2]))(Pstars)
# save(util1_B_cont_v_col1, util1_B_cont_v_col2, file = 'data/util1_B_cont_v_col.rda')

load('data/util1_A_cont_v_col.rda')
load('data/util1_B_cont_v_col.rda')

par(mar = c(2.6,2.6,0,0), oma = c(1,1,0,0), mgp = c(1.1, 0.2, 0), 
    cex.axis=1, cex = 2.7, tck = -0.02, 
    bty = 'l',
    font.main = 1)

for (i in 1:2){
Q = Qs[i]

util1_A_stop_v = util1_A_stop(Pstars) + Q
util1_B_stop_v = util1_B_stop(rep(P1, length(Pstars))) + Q

util1_A_cont_v = get(paste0('util1_A_cont_v_col',i))
util1_B_cont_v = get(paste0('util1_B_cont_v_col',i))

plot(Pstars, util1_A_stop_v, type = 'l', lwd = 5,
     xlim = axis_lim,
     ylim = axis_lim,
     xlab = 'Exchange rate $P_*$',
     ylab = "Alice's utility at $t_1$, $U_{t_1}^\\mathcal{A}$")
lines(Pstars, util1_A_cont_v, col = col_cont, lwd = 5)

p0boundsind = which(diff(util1_A_stop_v < util1_A_cont_v)!=0) # %>% .[length(.)-(1:0)] #take the last two

points(
  sapply(p0boundsind, function(x) mean(Pstars[x+(0:1)])),
  sapply(p0boundsind, function(x) mean(util1_A_stop_v[x+(0:1)])),
  pch = '.', cex = 5
  # col = c('red','blue'), pch = 15, cex = 1
  )

lines(Pstars, util1_B_cont_v, col = col_cont, lty = 2, lwd = 5)
lines(Pstars, util1_B_stop_v, lty = 2, lwd = 5)

p0boundsindB = which(diff(util1_B_stop_v < util1_B_cont_v)!=0)

if(length(p0boundsindB)>0){
  points(
  c(mean(Pstars[p0boundsindB[1]+(0:1)]), mean(Pstars[p0boundsindB[2]+(0:1)])),
  c(mean(util1_B_stop_v[p0boundsindB[1]+(0:1)]), mean(util1_B_stop_v[p0boundsindB[2]+(0:1)])),
  pch = '.', cex = 5 
  # pch = 0, cex = 1, col = 'red'
  )
}

legend('bottomright', lty = ifelse(i==1,1,2),
       lwd = 5,
       seg.len = 1,
       bty = 'n',
       col = c(col_cont, 'black'),
       legend = paste0(
         '$U_{t_1,\\mathrm{c}}^\\mathcal{', ifelse(i==1,'A','B'),
         c('}(\\mathit{cont})$', '}(\\mathit{stop})$')
       )
       )

assign(paste0('feasible_Pstar',i), Pstars[which(util1_A_stop_v < util1_A_cont_v  & util1_B_stop_v < util1_B_cont_v)])
}

# legend('topleft', lty = 2,
#        lwd = 5,
#        seg.len = 1,
#        bty = 'n',
#        col = c(col_cont, 'black'),
#        legend = c(
#          '$U_{t_1,\\mathrm{c}}^\\mathcal{B}(\\mathit{cont})$',
#          '$U_{t_1,\\mathrm{c}}^\\mathcal{B}(\\mathit{stop})$'
#          )
#        )


save(feasible_Pstar1, feasible_Pstar2, file = 'data/feasible_Pstar.rda')
```

```{r success_plot_col, include = T, eval = F}
# reset everything
source('DefaultValues.R')

load('data/feasible_Pstar.rda')

# sucrate_v_col1 = mclapply(feasible_Pstar1, function(x) sucrate(x, alphaA = alphas[1], alphaB = alphas[2], colla = Qs[1])) %>% unlist
# sucrate_v_col2 = mclapply(feasible_Pstar2, function(x) sucrate(x, alphaA = alphas[1], alphaB = alphas[2], colla = Qs[2])) %>% unlist
# 
# save(sucrate_v_col1, sucrate_v_col2, file = 'data/sucrate_v_col.rda')

load('data/sucrate_v.rda')
load('data/sucrate_v_col.rda')

par(
  mar = c(2.6, 2.6, 0, 0),
  oma = c(1, 1, 0, 0),
  mgp = c(1.1, 0.2, 0),
  cex.axis = 1,
  cex = 2.7,
  tck = -0.02,
  bty = 'l',
  font.main = 1
)

plot(
  Pstars,
  sucrate_v,
  type = 'l',
  xlab = 'Exchange rate $P_*$',
  ylab = "Success rate $\\mathit{SR}$",
  xlim = axis_lim,
  ylim = c(0, 1.05),
  col = col_success[2],
  lwd = lwd_success
)
lines(
  feasible_Pstar1,
  sucrate_v_col1,
  col = col_success[3],
  lty = 3,
  lwd = lwd_success
)
lines(
  feasible_Pstar2,
  sucrate_v_col2,
  col = col_success[1],
  lty = 2,
  lwd = lwd_success
)

legend(
  'topleft',
  legend = c('0 (basic setup)', Qs),
  col = col_success[c(2, 3, 1)],
  lty = c(1, 3, 2),
  lwd = lwd_success,
  bty = 'n',
  title = 'Collateral $Q$'
)
```