Bandits
One step
To illustrate bandits, we consider a very simple DGP:
\(Y(0) \sim N(5,2^2)\)
\(Y(1) \sim
N(5.5,2^2)\)
This means that the optimal policy rule assigns everybody to
treatment one. Let’s see how UCB and Thompson sampling figure this out
on-the-fly. For a stream of 1000 observations.
if (!require("tidyverse")) install.packages("tidyverse", dependencies = TRUE); library(tidyverse)
if (!require("ggridges")) install.packages("ggridges", dependencies = TRUE); library(ggridges)
set.seed(1234)
# Define parameters
gamma0 = 5
gamma1 = 5.5
sd0 = 2
sd1 = 2
n = 1000
# Draw the potential outcomes of all individuals
Y0 = rnorm(n,gamma0,sd0)
Y1 = rnorm(n,gamma1,sd1)
hist(Y0)
hist(Y1)
As a kick-start, we assign the first four individuals “manually” to
control/treatment/control/treatment:
# Assign individual 1 to treatment zero
W = 0
# and observe the potential outcome under non-treatment
Y = Y0[1]
# Assign individual 2 to treatment one
W = c(W,1)
# and observe the potential outcome under treatment
Y = c(Y,Y1[2])
# Assign individual 3 to treatment zero
W = c(W,0)
# and observe the potential outcome under treatment
Y = c(Y,Y0[3])
# Assign individual 4 to treatment one
W = c(W,1)
# and observe the potential outcome under treatment
Y = c(Y,Y1[4])
# Print the realizes values
W
[1] 0 1 0 1
Y
[1] 2.585869 6.102933 7.168882 6.770741
Now calculate the means in each treatment arm:
gamma = c(mean(Y[W==0]), mean(Y[W==1]))
gamma
[1] 4.877375 6.436837
And the standard errors of the means:
se = c(sd(Y[W==0]) / sqrt(sum(W==0)),
sd(Y[W==1]) / sqrt(sum(W==1)))
se
[1] 2.291507 0.333904
Upper Confidence Bound (UCB)
For \(i=5\), we implement now
explicitly UCB \[W_{i+1} = argmax_w
(\hat{\gamma}_{i,w} + \alpha \hat{\sigma}_{i,w})\] with
alpha = 2
:
# Calculate upper confidence bound
alpha = 2
upper_ci = gamma + alpha * se
upper_ci
[1] 9.460389 7.104645
# Choose the treatment with the highest bound
which.max(upper_ci)-1
[1] 0
Let’s visualize the decision.
plot_ucb = function(gamma, se, alpha, unit = FALSE) {
# Calculate UCB and find the arm with the highest UCB
ucb <- gamma + alpha * se
highest_ucb_arm <- which.max(ucb)
# Data frame for plotting
data <- data.frame(
Arm = factor(1:(length(gamma))),
Estimate = gamma,
UCB = ucb,
Highlight = ifelse(1:length(gamma) == highest_ucb_arm, "forestgreen", "orange")
)
# Plot
g = ggplot(data, aes(x = Arm)) +
geom_errorbar(aes(ymin = Estimate, ymax = UCB, color = Highlight), width = 0.01, linewidth = 1) +
geom_segment(aes(x = as.numeric(Arm)-0.05, y = UCB, xend = as.numeric(Arm)+0.05, yend = UCB, color = Highlight), linewidth = 1) +
geom_point(aes(y = Estimate), size = 4, color = "black") +
scale_color_identity() + geom_hline(yintercept = 0) +
labs(y = "Estimate / Upper Confidence Bound", x = "Arm",
title = "UCB Sampling for Multi-Armed Bandits") +
theme_minimal() + ylim(0,10) + theme(plot.title = element_text(hjust = 0.5))
if (is.numeric(unit)) g = g + labs(title = paste("UCB Sampling of treatment for unit",unit+1))
print(g)
}
plot_ucb(gamma,se,2)
![](data:image/png;base64,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)
Thompson sampling
Thompson sampling draws a value for each arm from an arm specific
normal distribution \(\tilde{\gamma}_{i,w}
\sim N(\hat{\gamma}_{i,w},\alpha^2\hat{\sigma}^2_{i,w})\) and
sets \[W_{i+1} = argmax_w
\tilde{\gamma}_{i,w} \]
# Thompson
draws = c(rnorm(1,gamma[1],alpha*se[1]),
rnorm(1,gamma[2],alpha*se[2]))
draws
[1] 0.4143512 6.3703030
which.max(draws)-1
[1] 1
Let’s visualize the decision.
plot_ts = function(gamma, se, alpha, gamma_tilde=draws, unit = FALSE) {
# This is probably the most complicated and unelegant plot I ever created, but it makes the point
# Let me know if you have an alternative to plot it more elegantly
highest_arm <- which.max(gamma_tilde)
# Data frame for plotting
data <- data.frame(
Arm = 0:1,
Estimate = gamma,
Draw = gamma_tilde,
se = se * alpha
)
plot_data <- data %>%
group_by(Arm) %>%
slice(rep(1:n(), each = 10000)) %>%
mutate(
y = rnorm(n(), mean = Estimate, sd = se),
) %>%
mutate(
dens = dnorm(y,mean = Estimate, sd = se),
) %>%
ungroup()
# Plot
g = ggplot(plot_data, aes(y=Arm)) +
geom_point(data=data, aes(y=Arm, x = Draw,color=Draw), size = 5, shape = 4, stroke=1) +
scale_color_gradient(low = "orange", high = "forestgreen") +
geom_point(data=data, aes(x=Estimate), size = 3, color = "black") +
geom_density_ridges(data = plot_data[plot_data$Arm=="0",], aes(y=Arm,x = y, height = dens),
alpha=0.7, position = position_nudge(y = 0), stat = "identity") +
geom_density_ridges(data = plot_data[plot_data$Arm=="1",], aes(y=Arm,x = y, height = dens),
alpha=0.7, position = position_nudge(y = 0), stat = "identity") +
labs(y = "Arm / Normal distribution", x = "Estimate",
title = "Thompson Sampling") +
theme_minimal() + scale_y_continuous(breaks = c(0, 1), labels = c("0", "1")) +
# # xlim(-10,20) +
geom_vline(xintercept = 0) +
theme(legend.position="none", plot.title = element_text(hjust = 0.5)) +
coord_flip(ylim = c(0, 3),xlim = c(-10, 20))
if (is.numeric(unit)) g = g + labs(title = paste("Thompson Sampling for unit",unit+1))
print(g)
}
plot_ts(gamma,se,2,draws)
![](data:image/png;base64,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)
Note that with this very seed, UCB and Thompson sampling do not take
the same choice \(\Rightarrow\) their
treatment sequences differ. Note that the decision of UCB is
deterministic, while it is random for Thompson sampling.
Dynamic treatment assignment (UCB)
First, proceed with UCB and define a function that takes
and returns a treatment assignment for the next individual:
ucb = function(Y, W, alpha, plot = FALSE){
gamma = c(mean(Y[W==0]), mean(Y[W==1]))
se = c(sd(Y[W==0]) / sqrt(sum(W==0)), sd(Y[W==1]) / sqrt(sum(W==1)))
assign = which.max(gamma + alpha * se)
if (isTRUE(plot)) plot_ucb(gamma,se,2, unit = length(Y))
return(assign-1)
}
Now we run UCB until \(i=1000\) and
plot the evolution:
# Store to use it for Thompson below
Wstart = W
Ystart = Y
plot_grid = c(5:10,20,50,100,200,500,1000)
for (i in 5:1000) {
# Assign unit to treatment (and sometimes plot)
if (i %in% plot_grid) Wi = ucb(Y,W,alpha,plot = TRUE)
else Wi = ucb(Y,W,alpha,plot = FALSE)
W = c(W,Wi)
# and observe the potential outcome under treatment
Y = c(Y,(1-Wi) * Y0[i] + Wi * Y1[i])
}
![](data:image/png;base64,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)
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)
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)
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)
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)
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)
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jdoa4P6SAoAAoEUTeve1z86WhAUBAUAAWCqOWaBUEBUCCIunGN8XQbckfXGE8hKAAKBFG3zY0Mfhzq/mspBAVAgSBq3VfUgqAAKBBE3TyWvf/ioQ5m5xAUAAWCqHWOlgVBAVAgiLru2v5/b8Xxwe9udTIYOAgKgAJB1DXXdobZpX103kgKQQFQIIi6Med/zX3VeSMZBAVAgSBq3S/bgqAAKBBEXV+vZQfWtF7LICgACgRR1/bXNqOBu1/2UPtrKQQFQIEg6vo85LOhO5Y9uNbJYOAgKAAKBFEf5X7ZfBAUAAWCqHUs24KgACgQRF137fmlMylnNQ8ZUxQABYKoG3MjUSTXphAUAAWCqBvHspc7GggkBAVAgSDq9mPZIoOgACgQRN1+LFtkEBQABYKo6+ceR0ua8C8hKAAKBFHXtiFX3YFszY0UEBQABYKo666dOaM5/xKCAqBAELWOZVsQFAAFgqh17wwLggJ0w7/OQddj/Ojo3hkWcu2YkGu6d0YNuXZMyDXdO6OGXAtFH92qo+v5l/zvHHQ9xr4i17iQa93B59qH3TvjdU+Yx7Wux9hXEte6HsIxMcs13TujgtZjweBbr+neGRXkWjAYXdO9MzzkWjA4XRNT5FowyFwrTzzWuccZci0YjK59EkWn3B+5Fsu1gJC55ti76Cb7J+PFR90MBwu5Fgw+13Q9/wpyLRh8ruk+NRXkWjD4XJvkJ/gf7byR3iLXgsHnWiLZ4OrLF/d1n5oUuRYMQtd0nxofuRYMRtfieP/BA503kiHXgsHpmpgi14JB5trB6tnHumadj1wLhlwjR64Fg8y1yXdXNNFfQa4Fg8y1g5WFxzqG7SPXgkHn2uAPw8F/PUj5Was4uRYQMtficVSi9Vss1wLC5lq8/5PWaz5yLRh0rsWTHzQ94iHXgsHnmqgg14LB6NrzSzq+NkWuBYPQtc0oiuRagVwLBp9rk/VouaOBICLXgsHnmj6PXUGuBYPPteJ6IyJFrgWDz7V4J1rSpP8UuRYMPtcOVt2HsjU3kiPXgsHomj5T4yHXgsHnmqgg14LB6drk5UvtsmXItVCk57t3PYjjRvdfM5BrYZh+uqTrgRwvjfuKRuce/PR9pKl/h1wLgvdRrq6Hcqy0nzeyGS11Mhow5FoIoohENl3P30CuhUCuyTW5FoQoYpFt1jak7p0Ry7UgsLqWzY080NxIhlwLAK1rmvP3kWsB4HVNx7I95FoAmF0TU+RaCFhUq7n2/Hdu+nE8+KqbwaAh10LA6dq9bE7k2VCHslPkWhBIVKu4thMtPEoXdNmRDLkWBg7VKvepWZ/eJbvl3vRvvx6NRudfZd9slIt9Rq6FgsC0+n1qnjQXC954cm2ff/XuzoVjH1vnyLVwvO56AMfP3K5tl24drt1M3Pvi6bEPrmvkWji4XPMuorU7bGxDbtyYLr795m4uXM+Ra+Hgci3enM6IND9Tc7j2p9EoX5WlqzTPtdd9JXGt6yGIk067awcruWybzduvvf06Wa+9/eNTz7Ubcc+h2GFHgWy95s48js5eXo1mnXqcr8vq67V+QjIRjQKba/Hk/idJuc7emnFCZO4Xxf4aywFWFOhcm022Lvv2rltmmIekOXEIBbk25d2dm+Uxto3+H1+Ta4GRayWHa+m5Iu/uuCmRjdGISDXJFgC5xopcC41cY0WuhUausSLXQkPn2mT9tK5V55BroaFzTdeFLJBqgaFzLd6MfqsL+zjkWmDoXJv8sKr7imZItbDQuab7ipZItaDQuSZ8ZFpAOF17+Wv4YYBCUAAUCKKuu7b3XRQtPP5SF4hMISgACgRRN+6dMfj9cOGX8opa3BAUAAWCqGuujaPr7hhby/VGKCEoAAoEUTePZRd/OhoQFAQFQIEgarlmQVAAFAiibtxXNN2GbLnuMSUEBUCBIOq2uZHBj0PdVzSFoAAoEESt+4paEBQABYKom8ey91881MHsHIICoEAQdct5IxPtqhUQFAAFgqgb542sum3Ic486GQwcBAVAgSDqxtxItHj58sVZFz5mg6AAKBBE3ZjzT6/o37x3BicEBUCBIOr2ayDoWHYGQQFQIIi6vl6Taz4EBUCBIOra/trOcMnNQm7qPP8UggKgQBB1/Xojl6LBp5fTCZLLl6/QT/4TFAAFgqjr+2v+5aO0HUlQABQIotb1NCwICoACQdQ6b8SCoAAoEESt80YsCAqAAkHUOm/EgqAAKBBErfNGLAgKgAJB1DpvxIKgACgQRK3zRiwICoACQdQ6b8SCoAAoEESt80YsCAqAAkHUOm/EgqAAKBBErfNGLAgKgAJB1HLNgqAAKBBE7bk2+eFyxh9+pt9RyyEoAAoEUXuu+Ttry92NCAmCAqBAEHXbNuT+s6Hm/FMICoACQdTt+2s7RzpH67UQYgbvc213qPl+B8GbLQoEUbe7pnO0MggKgAJB1HLNgqAAKBBEPWt/TfdfcxAUAAWCqDUPaUFQABQIotbxNQuCAqBAELXOG7EgKAAKBFHrfEgLggKgQBC1XLMgKAAKBFHLNQuCAqBAELVcsyAoAAoEUcs1C4ICoEAQtVyzICgACgRR+8fXvjz3sLNxYEJQABQIovbXa89Xo2jxzzq2VkJQABQIoq5tQ+79OIwGVx52MhRACAqAAkHUzf21iVu9nb2l1VtMUQAUCKJunxt58eNv9JmamKIAKBBErXlIC4ICoEAQtVyzICgACgRRyzULggKgQBC1XLMgKAAKBFHLNQuCAqBAEHWLay9/DT8MUAgKgAJB1HXX9r6LooXHX37VyWDgICgACgRR11zbHQ5+P1z4ZV3X9kkhKAAKBFHXXBtH1921IXeHumadg6AAKBBEXb+v6MKT4k9HA4KCoAAoEEQt1ywICoACQdS1e9OvZ9uQuu5xBkEBUCCIum1uZPDjcHC7m+GAQVAAFAiirs/57150lz1elGopBAVAgSDq5rHs/RcPdTA7h6AAKBBEXdtf++FKtp92/9+0vxZTFAAFgqib85Du667mIVMICoACQdQz7lOjecgUggKgQBC1v177v8uXhoNP01vVXNUum4OgACgQRF3bhlw9q21HD4ICoEAQtT6/ZkFQABQIoq679nz1jGOouREHQQFQIIi65tpmMTdyTnMjMUUBUCCIun4+5OAv64Pbeyv6/FoKQQFQIIi6eXxtnHimc48zCAqAAkHUTdd2ouVYn6nJICgACgRRN7Yhb+8Ol+RaDkEBUCCIujY3kmw8Pl4ZXP0PbUOmEBQABYKo63P+z37zZHfY9qGaw7XR6Iun2fLbr0ej0flXAYbXMQQFQIEg6tZj2S1XiDxcuxDHG7lfbxg8cxAUAAWCqOc9b+TN53eT9dk3d9Nvti8c34CgICgACgRRzzhv5EzreZGFaxs3jn1cGBAUAAWCqBvXh4yi2a7l25CHa38q990SXgshZjDLtYOV6NpsL7c/z1Zrb79O1mtv//h09iP7AsGbLQoEUbd/LruV7dFN77vDtZszH9kbCAqAAkHU9W3I2Rer28jXajlyTXxMCKKuuTa5t3jrpaMx679d7qC9cYuH396tP6R/EBQABYKo6659n3+mpr4tWUxBOt7duUlyjI2gACgQRN24xvjinx84fq6do7U9Srn57s6N7BwSBtUYCoACQdRHmBshhKAAKBBEXV+vyTUfggKgQBB1494Zy92MAxOCAqBAEHX9GuOXokF6gcgr+kxNTFEAFAiibpw3MmMekhOCAqBAELWuD2lBUAAUCKKWaxYEBUCBIGr/3hmrZx/nn6iZcZ4/HQQFQIEgarlmQVAAFAii1jakBUEBUCCIWvcVtSAoAAoEUeu+ohYEBUCBIGrdV9SCoAAoEESt+4paEBQABYKodV9RC4ICoEAQdes85Aut1TIICoACQdSN60P+7sneinbXcggKgAJB1I37irobsP3LvWipm+GAQVAAFAiibt4T6mAlun6wohWbg6AAKBBE3Xavw4UnuhZCBkEBUCCIuunaZrL9uDuUaw6CAqBAEHXjev7n3H3pN7W/lkJQABQIoq65dnAxij5LVm+LWq05CAqAAkHUjeNr+1txPHnZxVAAISgACgRR6zM1FgQFQIEgas+17AM1++6cEc1DZhAUAAWCqCvn+SeCZZbJtQyCAqBAELVcsyAoAAoEUcs1C4ICoEAQtVyzICgACgRRyzULggKgQBC1XLMgKAAKBFHLNQuCAqBAEPWsa/vINQdBAVAgiLpyLPtyie4J5SAoAAoEUescLQuCAqBAELVcsyAoAAoEUcs1C4ICoEAQtVyzICgACgRRyzULggKgQBC1XLMgKAAKBFHLNQuCAqBAELVcsyAoAAoEUcs1C4ICoEAQ9cdw7bUQYgYf17X+QvBmiwJB1HLNgqAAKBBELdcsCAqAAkHUcs2CoAAoEEQt1ywICoACQdRyzYKgACgQRC3XLAgKgAJB1HLNgqAAKBBELdcsCAqAAkHUcs2CoAAoEEQt1ywICoACQdRyzYKgACgQRC3XLAgKgAJB1HLNgqAAKBBELdcsCAqAAkHUcs2CoAAoEEQt1ywICoACQdRyzYKgACgQRC3XLAgKgAJB1HLNgqAAKBBELdcsCAqAAkHUcs2CoAAoEEQt1ywICoACQdRyzYKgACgQRC3XLAgKgAJB1HLNgqAAKBBELdcsCAqAAkHUcs2CoAAoEEQt1ywICoACQdRyzYKgACgQRC3XLAgKgAJB1HLNgqAAKBBELdcsCAqAAkHUcs2CoAAoEEQt1ywICoACQdRyzYKgACgQRC3XLAgKgAJB1HLNgqAAKBBELdcsCAqAAkHUcs2CoAAoEEQt1ywICoACQdRyzYKgACgQRC3XLAgKgAJB1HLNgqAAKBBELdcsCAqAAkHU87u2MRqdf9Wy3GcICoACQdRzu7Z9/tW7Oxeay72GoAAoEEQ9r2uHazfj+M0XT+vL/YagACgQRD2va2+/uZtLVl3uNwQFQIEg6nldS1djuV/+cvocQogZfLBrN+rL/YbgzRYFgqg/xnqtvxAUAAWCqLW/ZkFQABQIotY8pAVBAVAgiHru42sb3jG1DR1fEx8ZgqiPdN5Iote7Ozemy/2HoAAoEESt8yEtCAqAAkHUcs2CoAAoEEQt1ywICoACQdRyzYKgACgQRC3XLAgKgAJB1HLNgqAAKBBELdcsCAqAAkHUcs2CoAAoEEQt1ywICoACQdRyzYKgACgQRC3XLAgKgAJB1HLNgqAAKBBELdcsCAqAAkHUcs2CoAAoEEQt14QIg1wTIgxyTYgwyDUhwiDXhAiDXBMiDHJNiDDINSHCINeECINcEyIMcs3g8Nu7XQ+Bg8O10aj/16yXawYbn8u1EByuXUgvXN/1OI4ZuTaT5M1WrgXhjcs5vftRr5FrM3lzof//+4Hof9hyzaD///uB0DYkNXItHNv9316XawZyLRjbo/7fplauGci1UFDM+Mo1A7kWiO3+H1yL5ZqJXAsDSc5yzYCkA52zPUrp+x6bXBMiDHJNiDDINSHCINeECINcEyIMck2IMMg1IcIg14QIg1zrBTvR9a6HIN6HXOsF42ip6yGI9yHX+sDu8MzgdteDEO9BrvWBzYX/GS4nXyfrS5vRwk/D5WfDwVfxzjA6t9X10MQUudYDDlaWJuunt5xrp4bR6X8OPx1G0eA/k/9o0xIIudYD3MxIOjsyWY+W3RZldC1+FiWLu8PTWrHBINd6wHjhiVu3OdeSpcSwJbeuSzTL1nYCA7l28tl1G4tRlGiWubXr9t1S9+KxXMNBrp18NqOMZbkGjVw78eQbim6jUa4hI9dOPDtuPiRhPLgt15CRayeecX4YeydakmvIyLWTTjrr6JisL/wi14CRa0KEQa4JEQa5JkQY5JoQYZBrQoRBrgkRBrkmRBjkmhBhkGtChEGuCREGuSZEGOSaEGGQa0KE4f8BotSyegPDOvcAAAAASUVORK5CYII=)
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)
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)
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)
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)
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)
additionally plot the share being assigned to the optimal policy
share = cumsum(W) / 1:n
plot(1:n,share)
![](data:image/png;base64,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)
and the regret
regret = cumsum(gamma1 - (W * gamma1 + (1-W) * gamma0) ) / 1:n
plot(1:n,regret)
![](data:image/png;base64,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)
We observe that as observations flow in, UCB figures out that
treatment one is dominant and starts to solely assign it. Such that at
the end of the 1000 observations regret is negligible. Note that the
regret of a 50:50 experiment would have been 0.25.
Dynamic treatment assignment (Thompson)
Write a function similar to ucb
above:
thompson = function(Y, W, alpha, plot = FALSE){
gamma = c(mean(Y[W==0]), mean(Y[W==1]))
se = c(sd(Y[W==0]) / sqrt(sum(W==0)), sd(Y[W==1]) / sqrt(sum(W==1)))
draws = c(rnorm(1, gamma[1], alpha * se[1]),
rnorm(1, gamma[2], alpha * se[2]))
assign = which.max(draws)
if (isTRUE(plot)) plot_ts(gamma,se,2,draws,unit = length(Y))
return(assign-1)
}
Now we run Thompson until \(i=1000\)
and plot the evolution:
# Restore starting points
W = Wstart
Y = Ystart
plot_grid = c(5:10,20,50,100,200,500,1000)
for (i in 5:1000) {
# Assign unit to treatment (and sometimes plot)
if (i %in% plot_grid) Wi = thompson(Y,W,alpha,plot = TRUE)
else Wi = thompson(Y,W,alpha,plot = FALSE)
W = c(W,Wi)
# and observe the potential outcome under treatment
Y = c(Y,(1-Wi) * Y0[i] + Wi * Y1[i])
}
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)
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)
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)
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)
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)
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)
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)
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)
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qRTunzm2xjy7fmPrWXobUuWIlyZ1w38mp710X2u/bdpnfhgslau9lDarTmggXDLP19SGNeqtYKdJ1eobWV+JpfevOKkw6O4vk1c73t4bfSa7/19qZa6/5LLLHS320/ycbG1Ehp87xagtLaXz69fGsdfElzUfpue0I2NqaHyn5bf6zup5+SW+k6+PZbx1/k0mt/kfCy1tr4EhpDa93T7HBt5vHa6Z31j3+wLDvr/1McW3Ze91u91qapFx6trSAN66hdTzOKrX0UL27pfckwtmTcPewjtS7X/hnf+dzX3AP6rbXcxVJ1tEvpb4yv/+m/LF8ufY1WEJtJxt3NPlJzNO7KK1TkGq1WeEmz6IQO50byrfWRmutxlw6WCxf6OLMWscBvMLWUBtdDzqO4XWvU2rOzMVlrN7994Avh1xzE5tK3EHPWHrTDrNYm4+Hm4t0WybfW4Y/ymfRz7GK3aW2cyG6+ZQl//LW3+hrCdm0exb8x/rYZhH8g8t0WfqfGy9bC3cf/qXeBVv2EakNrq0vpupGYNo7bfGwtSMxEcyPl2Nrq6SxobXXpdNona62HUd/f73LF5+/j2DpNqwStrS601gJpTBeiXxfl70NCBfn3zrj62lfxb9S82tJ1/klrrgb9NB23Vt5gJaklf/c4/dgdtLa6dNva5CfbWh/n1iztt3b20VaW1oP4Tvex0drq0u0ujr1Q60Ev14xYWmmtwWRGFpYpf8oRtLa6dPq+ouE1kQ96uRbSslhrbcwV+jjiaW0eXb6vaHStvx3x/cRWs7UOJuJ9HPG0No8O31c0/rWacMSv1PWQnOV6ucwqdPe+oslvsEUjvs/r/IttdT8N7+OIp7V5dPa+oklqye9lu4zNeVtFfBzxtDaPynH4fXtbtexvIHQaW2VcPo47H80qlP4+5G/vH2+3cLjW+d8bqbHp8nHc+WhWofS+ovYN2P7ujtlozdDiGeWGO4Y+jjsfzSqU3xPqZNvcONlenfepWfzAy8dx56NZhar3Oly73/v7r7UyseHjuPPRrEK5tVGw/zge9tNau1OHPo47H80qlP6e/0X7vvQjx8dr3czO+zjufDSrUBjdJzvGXAo2b+vt/Tmts1vr8gyYj+POR7MKpVH+8+FkcvpDi4azr5RqUVTCx3Hno1mFXt7r0NHFHD6OOx/NKuSGfPQLNT/ba0Y6nId0eM2Uj+POR7MKU9f5B4FFlXXVmtuLE30cdz6aVXDYmvOLgH0cdz6aVXDWWg+X2/s47nw0q+CotV7efsTHceejWQU3rfXzTj8+jjsfzSo4aa2nN9Xycdz5aFbBRWt9vX2dj+POR7MKs/62T4ut9fZWkT6OOx/NKkydy34no433hIp40Nu7svo47nw0q+DgGi1awwwTWsOsb1ah+xB6S83LceejWQVaw6xuVoHWMKubVeDvbWNWN6tAa5jVzSrQGmZ1swq0hlndrAKtYVY3q0BrmNXNKtAaZnWzCrSGWd2sAq1hVjerQGuY1c0q0BpmdbMKtIZZ3awCrWFWN6tAa5jVzSrQGmZ1swq0hlndrAKtYVY3q0BrmNXNKtAaZnWzCrSGWd2sAq1hVjerQGuY1c0q0BpmdbMKtIZZ3axCw9YOtrYuP232FB/XPmYo06y1J5ef/vrJlWYGH9c+ZijTqLUXt25PJs/f+qaRwce1jxnKNGrtl/c/jYNrgI9rHzOUadRauEnLtWYA4GyWa+2DJs/x8ictZiiz1HatFj6ufcxQhuM1zOpmFZiHxKxuVqHZ+bUDzq9hXjmzCs2vG2mYmpdrHzOU4XpIzOpmFWgNs7pZBVrDrG5WgdYwq5tVoDXM6mYVaA2zulkFWsOsblaB1jCrm1WgNczqZhVoDbO6WQVaw6xuVoHWMKubVaA1zOpmFWgNs7pZBVrDrG5WgdYwq5tVoDXM6mYVaA2zulkFWsOsblaB1jCrm1WgNczqZhVoDbO6WQVaw6xuVoHWMKubVaA1zOpmFWgNs7pZBVrDrG5WgdYwq5tVoDXM6mYVaA2zulkFWsOsblaB1jCrm1WgNczqZhVoDbO6WQVaw6xuVoHWMKubVaA1zOpmFWgNs7pZBVrDrG5WgdYwq5tVoDXM6mYVaA2zulkFWsOsblaB1jCrm1WgNczqZhVoDbO6WQVaw6xuVoHWMKubVaA1zOpmFWgNs7pZBVrDrG5WgdYwq5tVoDXM6mYVaA2zulkFWsOsblaB1jCrm1WgNczqZhVoDbO6WQVaw6xuVoHWMKubVaA1zOpmFZZq7RkAnE1LrdULsnMDZr/NKtAaZnWzCrSGWd2sAq1hVjerQGuY1c0q0BpmdbMKtIZZ3awCrWFWN6tAa5jVzSrQGmZ1swq0hlndrAKtYVY3q0BrmNXNKtAaZnWzCrSGWd2sAq1hVjerQGuY1c0q0BpmdbMKtIZZ3awCrWFWN6tAa5jVzSrQGmZ1swq0hlndrAKtYVY3q0BrmNXNKtAaZnWzCrSGWd2sAq1hVjerQGuY1c0q0BpmdbMKtIZZ3awCrWFWN6tAa5jVzSrQGmZ1swq0hlndrAKtYVY3q0BrmNXNKtAaZnWzCrSGWd2sAq1hVjerQGuY1c0q0BpmdbMKtIZZ3awCrWFWN6tAa5jVzSrQGmZ1swq0hlndrAKtYVY3q0BrmNXNKtAaZnWzCrSGWd2sAq1hVjerQGuY1c0q0BpmdbMKtIZZ3awCrWFWN6tAa5jVzSrQGmZ1swq0hlndrAKtYVY3q0BrmNXNKtAaZnWzCnVbe/H7T+0/B1tbl582M/i49jFDmbqtHbxpW3ty+emvn1xpZvBx7WOGMvVae3Fry7b24tbtyeT5W980Mvi49jFDmXqtPb/yy/tBa+F/wuAa4OPaxwxl6u5DhpmFm7Rca88A4GyWa+2D+ilP/PxJixnKLLVdq4WPax8zlJnX2pOtrS27HeN4DfOqmlVotF1jHhLzCppVaNTa5IDza5hXzqxCs9bsdSMNU/Ny7WOGMlwPiVndrAKtYVY3q0BrmNXNKtAaZnWzCrSGWd2sAq1hVjerQGuY1c0q0BpmdbMKtIZZ3awCrWFWN6tAa5jVzSrQGmZ1swq0hlndrAKtYVY3q9B9awBgoTUAN9AagBtoDcANtAbgBloDcAOtAbiB1gDcQGsAbqA1ADd03doCb/qrT/yGxwB5Om5tkTf91Sd6w2OAKbptbaE321AnfsNjgGm6bW2hN5FSJ37DY4Bpum1toTdH1IfWoAInrTV70199aA0qYLvWAbQGFXC81gG0BhUwD9kBtAYVdHx+bZE3/dWH1qACB9eNeJcarUEVXA8J4AZaA3ADrQG4gdYA3EBrAG6gNQA30BqAG2gNwA205pojc2PuQ84fhv+OzGb478n2Zu3Fn+7GT04+3Ch8avzb+7mHnmxvlJdhH1JYDiwPrblm31SM7sJD4keMzGDP/ttqa6O1ua3Zh9Ba69CaY8bDV6OAZnOyHW/5RsaEA37J1qbJWotcM1qD1qE1x4zW/ntoywkiGJm1L4abj4aD9yZHQ3MxTeQoGeqjtX8L9yKj1o53jFm/N+uZj14xJvgw39qjoVm/m2zXTj8K7r8e3DbGbEaLuBtt14IlRIu1TwzuSR5iP8yc5794JVw+LAytuSUY28mgPjc05/8yfH0YNPAfwX+yXctRcmu09vWu3QiGrR3Zx9idyspnjkzIjVxr4afWd6LWwoCC+5OQ7CJ+DFs7N0wWW24t50yWDwtDa26xMyPh7EgwejftHqW5PnkUDO3gVtLI6W6yxxjsy4Wftq0Fj79u+wmzKT3zdHftnv1c7uDsZHvw8eR410SfGg8vHYb3xwdjdhHR8Zp54/D0M7ORtZYdr005Nw6DW3MPNWE2tOaW/WAYh8dIQR33ozqCj+0+Xro9Gv99cjxnx7ydi7SthY8Mnj/Ym/HMH77449DkWhuHu6pJfuPhuX+5my40WkTUWlTY2v2K1orOyoM7qAutOWUc7pSZdJ4vDCIawvtJa0fpAVc05gd7UWvh/tuRHfcVz4wXnGstOreQzkPu27sv3stPMkathRvRUbLYQmvTzvJECzSA1pwSH1aZzTNa208nHUfRBuz83+a2FuwKvv6nL/+2Pbu1yeOdKHJa6wtac0kcgt1xm9laOuOfzL2PzD8W9iHLz4yqmDpei/YhT/L5ff+HoL9iaxuTeId2P91NrNqHpLWloTWXHMUXgswoJnpMdnIras3OAU7PjVS1tnFoJ+iTeyfh3Eg4xR+1dmTeCI7sPhvs2U3YVGt22XZuJNjLfC96fPqQopPWloLWXLIfn8Y+MhszWxtl4zk+pxwci03P+ZefmczJ51oLJOU5//N2MjE5eRa1Zk8d2H3L7PHZQwpOWlsKWnNIvE8W7rN9PaO1bMY/u34jui5ynJ1XLj8zPOn88X4ynRgSHKGtfz11LvtScNfJjtn4MX+8FgT1m3v28d8NzcUfbU7ZQ6adtLYUtAbgBloDcAOtAbiB1gDcQGsAbqA1ADfQGoAbaA3ADbQG4AZaA3ADrQG4gdYA3EBrAG74f5Jc3NsJSHXiAAAAAElFTkSuQmCC)
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)
![](data:image/png;base64,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)
additionally plot the share being assigned to the optimal policy
share = cumsum(W) / 1:n
plot(1:n,share)
![](data:image/png;base64,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)
and the regret
regret = cumsum(gamma1 - (W * gamma1 + (1-W) * gamma0) ) / 1:n
plot(1:n,regret)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAA2sAAAIcCAMAAABfO8ZvAAAAzFBMVEUAAAAAADoAAGYAOjoAOmYAOpAAZrY6AAA6OgA6Ojo6OmY6Zjo6ZmY6ZpA6ZrY6kLY6kNtmAABmOgBmOjpmOmZmZjpmZmZmZpBmkJBmkLZmkNtmtpBmtttmtv+QOgCQZjqQZmaQkGaQkLaQtraQttuQ2/+2ZgC2Zjq2kDq2kGa2tma2tpC2tra2ttu229u22/+2/9u2///bkDrbkGbbtmbbtpDbtrbbttvb25Db27bb29vb2//b////tmb/25D/27b/29v//7b//9v///8bQmi6AAAACXBIWXMAABJ0AAASdAHeZh94AAAUYElEQVR4nO3dDXvayKGG4cGOI+OepMZuvc2eE5J2E3vTOqXxWaddYhsj/v9/KggJJAMSSOKdD577ujZONtgzxjyRNBJgJgAUjO0JAAeC1gANWgM0aA3QoDVAg9YADVoDNGgN0KA1QIPWAA1aAzRoDdCgNUCD1gANWgM0aA3QoDVAg9YADVoDNGgN0KA1QIPWAA1aAzRoDdCgNUCD1gANWgM0aA3QoDVAg9YADVoDNGgN0KA1QIPWAA1aAzRoDdCgNUCD1gANWgM0aA3QoDVAg9YADVoDNGgN0KA1QIPWAA1aAzRoDdCgNUCD1gANWgM0aA3QoDVAg9YADVoDNGgN0KA1QIPWAA1aAzRoDdCgNUCD1gANWgM0aA3QoDVAg9YADVoDNGgN0KA1QIPWAA1aAzRoDdCgNUCD1gANWgM0aA3QoDVAg9YADVoDNGgN0KA1QIPWAA1aAzRoDdCgNUCD1gANWgM0aA3QoDVAg9YADVoDNGgN0KA1QIPWAA1aAzQatzb80cY0gODVbC3+/vVh+mH0wRjTeffQ6pSAINVsbdw7+jb71XT+58/GnJTGZoDwiFu7Nq+nlcV9c76HIQCHiVsb9zqfZ394iko3bLSG8Mhbm23cJouP7Q4BOEy+D5lt12gNB0bZmumcvft7NDtQi6eHbXsYAnCYrrX4/mN3thiT7Ema0s0arSFAutYSz9//+qdZa28GexsCcJO4tcZD1DpLATjAr9aS0KgNXvKsNdUEgNZZai3++FONc9mm/K8Bl1lqbc257C2uHKM1eMzWPuRz6fNqaA3h4XgN0FC3Fg+Hw+rnrrEOifBIW7u/TA/Hzj7VHYLza/CVsLW4b0zn7Orq6jKqfK5ozSEAdwlbuzNH2ZVZox7PFcWhEV573J8/nSbBc0VxcNTPX1v3h9aGABxmabv2WH7ARmsIj/R4rZMtP95HHK/h0CjX/G+MMcfdbvd0+vHtfoYAnCU9vzb6cJqcXjuueKoorSFAfl2jBfiL1gANWgM0aA3QoDVAg9YADVoDNGgN0KA1QIPWAA1aAzRoDdCgNUCD1gANWgM0aA3QoDVAg9YADVoDNGgN0KA1QIPWAA1aAzRoDdCgNUCD1gANWgM0aA3QoDVAg9YADVoDNGgN0KA1QIPWAA1aAzRoDdCgNUCD1gANWgM0aA3QoDVAg9YADSutPf/Y+xCAa2y0Nu4dfdvzEIBzdK3F329T/4g6v9x+fWh/CMBhutbGPZNXummjNYRHuA95f2HesF3DwVIer8U35mQw4XgNh0m7NvJ00fmZ1nCYxOuQ8a/m1YDWcIjka/6ji85faA0HSH9+Lf4SlS9CNh8CcJCNc9mjjz+VLUK2MQTgHK6HBDRoDdCw1FpcvhtJawiPpdbWLPsXruBqYQjALbb2IcufVkNrCA/Ha4CGurV4OBxWLPg3HQJwkrS1+8v0cOzs076GAFwlbC3uG9M5u7q6uoyMOSnduNEawiNs7c4cDdLfjnrmfB9DAO4SvgZCv/N58YenqHTDRmsIj/I1EHKn1CqeVkNrCI+l7dpj+QEbrSE80uO1Trb8eB9xvIZDo1zzvzHGHHe73dPpx7f7GQJwlvT82ujDaXJ67fjNoPyGtIbwcI0WoEFrgAatARq0BmjQGqBBa4AGrQEatAZo0BqgQWuABq0BGrQGaNAaoEFrgAatARq0BmjQGqBBa4AGrQEatAZo0BqgQWuABq0BGrQGaNAaoEFrgAatARq0BmjQGqDhXWvGECK85FlrSWjUBh/51ppqBkDb/GrNVPw94C5aAzRoDdDwqzWO1+Av31pjHRK+8qw1zq/BW961BniqYWvPP9LfxMM2ZrNuCCAMzVob946+FX/TDlpDeBq0Nrq9/UfU+eV25ldDa0CpBq2Neybn9fZfoHp/k9YQnib7kL8vt2u3g20+8/n263RzeDEt881D27MCHNfseC3++FN5NAU308he/atvXl1FFbuctIbwtLHm/2PdrVbdmc67L5Ex59NGr2e/tjorwHFNW4u/zLZR4z99rvy0uN+Z3uhxvkV7ik7KNoi0hvA0bC3um+NZa71OZWzz0wLjXhJZxTkCWkN4GrZ2Z86TbB6r1yHnPcbf/zZr7SmiNRyYhmsj0/3CpLVtzmVfm9fZfuP0eK20TVpDeJpfN7J1a+MLs7jKhHVIHJyWtmsVax3prb+cZa29KT8fR2sIT8Pjtev58Vrc3+W6kd2GAMLQsLXpfuFx1PlD1bnpJkMAYWh8fu0muRqy4pqrRkMAQWjhupHn4e7PXau4uIvWEJ6GayMf3tbboK1Zt8w/aYDWEJ6Wniu6s+fSiyhpDeFpvubf3lzWDgEEouHx2uji1S/DxJYX+8fTm1bvdtIawtN0H3JxhLXNzuT9ZXrjs0+tzwpwXNPnil5lqp80GveN6ZxNb3oZGVN+mQmtITzC14e8M0fZlVmjHs8VxaHRtVZYR+G5ojg4re1DXv3ln+WfVjg/wHNFcXBaWxuZvWxd6TFYYbv2WH7ARmsIT9PXQLjp/Pxj8vxbZH7+vV9+DDZ7bZ9s+fE+4ngNh6bxayC8Tz7OtlOVz2GbXaZ83O12T6cf37Y9K8BxLb6ef/UFW6MPp8nO5nHFU0VpDQGStrbPWQGOa/y87NeLj1u9DsLuQwBhaNjaU2SO393+9cLMXnekYm1kv7MCHNf0XHbyThjGvBpkr7Jqa1aA45pfNxL/fvu13VdAoDWESPjeGU2GALyne++M2kMAQdC9d0bdIYAw6N47o+4QQBiE751RbwggEML3zqg3BBAI5Xtn1BoCCATvnQFo8N4ZgAbvnQFoWHrvjB2HAPxn6b0zth8CCISt987YeoiXf8O72MBTfr13RhIatcFL6vfO2H2I4v83i18Bv0jfO6POEMX/bQofAY8I3zuj3hDF/01r8JbwvTOaD0Fr8JhXrXG8Bo951hrrkPCWX61xfg3+8q01wFe0BmjQGqBBa4AGrQEatAZo0BqgQWuABq0BGrQGaNAaoEFrgAatARq0BmjQGqBBa4AGrQEaytbim2737J/JbyteMJnWEB5ha+OL5AUMkncAoDUcHGFr1+ZkMBn1zewdSGkNB0fXWvba/zeztyClNRwcXWuLvK7NOa3h8ChbS9/TJu6b97SGgyM9Xjuf/2bcM/9Lazg0wtaeInMyD2y2IklrODDK82ujiyyw+IbWcGhsXTcS/7vsPaTKhuA1xuEn367RSkKjNnjIu9ZUUwBaZqm1+GPp+5BuHMJU3QBwlaXW1pxfM3lVY9MavGNrH/L5R60haA3e4ngN0FC3Fg+Hw7IjtaohWIeEr6St3V+mh2Nnn+oPwfk1+EnYWtw3pnN2dXV1GZnkSWztDwG4S9janTkapL8d9bLrkNsdAnCX/rmiiaeodMNGawiPheeKrvyhxhActME7lrZrj+UHbBVD5BYjiQ6+kB6vdbLlx/uo0fHaPDXDGQD4RLnmfzPt47jb7Z7OXrmuwRDLyDizDX9Iz6+NPpwmp9eO3wzKb1jdWvKRK7bgEd+u0Zr/bbo9Y8MGj/jY2nJZxKS/7H9GQFP+tjZfG0lXSFiOhPO8bG0ZWdoby5Fwn7etTeYbs+XSP3uScJunrWXbtMl8pYQlErjPw9ayJRFTfNmEwg4l4BzvWlssibwMLeuN2uAm/1qbZCfYclu3dEey8qWBAHt8a215hJYep23YugGu8b61F691R2twleet5Xci2bDBab61lj9ey2/M8v9x1RZc5F9rFeuQbNrgKO9aKzu/ViyN2uAUD1ubLDZsk8WF/mzZ4DxfW0sWQzat+KedERtc4mVri51EDtngDx9by7Zqi53HzfHtf6rAljxtLXc+LduXzJWWO+8GuMLD1gqlFRtb3bLtf67AdjxtLbucv9hVYZPGFf9wi4+tLWPKtnAvPxb2LwEn+NeayV7OJ7d5m2QHZ6vn2vY/WWArHra2WAyZTBapZYuSk9VDtv3PFtiGd60tEzPZc0SzLVv69/ni2ImEM/xsbTLJvUZk9r+LrxrJ6ggc429rE7M4aMsux1rsWE7yu5J7niuwHe9ae/H6dGlWL26SPwtAa3CDh60VLivecMtlbByxwRH+tTbJdh4rbrl8YQQu1oILfGwtd5OKE3Fb70pSI/bO39Yqn6G2zKxqkYQDOwh429oW26KXZ7U33Z5z31DwsrUti1htzeT/bvnLiw1go6kDG3jY2vavb7C+tRfHcWVFAu3xsbWtZ7Euo4qXAyI27It/rVWv9uduu/qCyNtp4XsAikJvbd2GjdhgQ9it1dqmURv2wr/WdjhemxAb3OFja1uvQ04atUZsaJWHre12RVUSzcuIiA16Xra24xdbfTG7tWfeiA175WVrtSrYchNGbdgTXWvx99u8rw91h6j7vhhb7i2ydcN+6Fob9woP4KNvdYfYaR2y8ImrFa3LqbJGoA7hPmTcNyctbNd2Or9W/Mx8Yfn/uemGFcGRIXahPF6L+53PzYeo31r+pe6qb1hVGxs97Ea6NjLunZRtzbYbokFry9i2uWFFcK3sYi4++8XXod8AadchH7vvmw9R+3htsohtuxvuZsNXWflalV+/GF7TmuEMD9f8665DZp+9tytOXo6y+Ya7f+3y4eADD1uT7WHVePibtW8psD+KuwEt8bI1FV0zTSwna/XOQhVaK2W7o21tnGj6Tdi+HzGx1lr88afa143ICZtRWfnucn/a4b7Y1z0eJkutjXsr1424/UNcebC2tcJhz5rvqqUvi7Vs7UM+/9j7EC3b8Mjax8M1FLZ/ZI7heG0nGx5NzR6RjT7bK9vdw/v8Adqkbi0eDofV1464fXeve/DUeuAtv4JZzfaAEtyO/gfdMmlr95fp3Xb2aV9DyKz89Dc8QFYeJtWPneVfFm9WeNy1/Uj2SHs/RC3tdf6mc3Z1dXUZGVN+YaSfd2fx8VDyqGjrAWPjge6QNu5CJWFrd+ZokP521DPn+xjCOv2jwOqj3RHCu7sJXWuFZ9Q8RaUbNl/uPUdsesTZLcAGp5dWdK0VTqmtOb/WwhBYwywvHmnrAb38yk6zd5+vZ2m79lh+wObc3RSetQ/INQ9Qk18h3fy13A6v5fuuJunxWidbfryPAj1ew5LtwNazeYdIPmXuZvqtHne73dPpx7f7GQIuMu5u96R3g+RTUqMPp8k3ePxmUH5DWgucWTnnuHoicvFXdiJsP0Su0YJXbLW3jLD+1CWf4uAQ8Jzt5oq2mnGNb3L3T3FwCITCdmYvbZ5oje+twf3izhAIju3IFjbNr8a31OgOcWUIBI7WnBkCB8OV2GgNB4TWbAwB0JpmCGCjvaRGa8AOaA2wZOvSaA1QoTVAg9YADVoDNGgN0KA1QIPWAA1HWwPCUyOE9tvyZPAlN6bhxiyYRl67s6A1V6bhxiyYRh6ttc2NabgxC6aRR2ttc2MabsyCaeTRWtvcmIYbs2AaebTWNjem4cYsmEYerbXNjWm4MQumkUdrbXNjGm7Mgmnk0Vrb3JiGG7NgGnm01jY3puHGLJhGHq21zY1puDELppFHa21zYxpuzIJp5AXUGnBAaA3QoDVAg9YADVoDNGgN0KA1QIPWAA1aAzRoDdCgNUCD1gANWgM0aA3QoDVAg9YADWutxV8i03n3YGXs0aUxnbcPL6ZhY0aP5r31Wfx/ZMwb63dGfBNZ/5mMe+8nmyfQeC7WWrtO3uzjtY2hn6Jk7JOH4jQszGjcm7dmcxbzAY++2Z1G3Lf/M5nOYd7a+gk0nout1p6io8H0l85n/dDTu/SPD5PRhTkvTMPGjKY/vtlP1+YsHs2rwWTUt31nzKeR/NNjaxqz3JPW1k+g+VxstXadfFuPNjZs414y6FP0ujANCzN6NN1kUIuziPvJg+cpmm5RbN4Zd+mA59bujd+iTjfby1g3geZzsdRa3E92Wsa9EztHbOnYuWlYmNE0+uQhZnMW6T88M1bvjGVrlqYR918N5pNYP4EW5mKptXTK6TdgxexfqNw0LMzo+uhb8tO1OYun6PzpYr42YvXOeIpm+5AX042srWkMs+DXT6CFuRxsa8nOk+WH1/uJ9dZmu7Hp2ojdf3h+my1Ydf7P6r0RamvzXZdra61dzw4NctOQzyjuT394aWv2ZvFozEmyNvLa6jQm8U2S/Fur90bW2roJtDCXA92uxdfJMa7Nf8rvZosSDmzX0sOQtv8R39Fdmvw527W2WW4t7pvktKnFn+v0QGniRGvJP9ftP7B2M2t9Yj35MFuzuw456s12IIvTUM/ozqSOvtlcAEzOfMzvCBeWQ6+tTiPMdUib59em99fsGPzlNMQzyrVm88TW9L74PEkfQlansdiuWf2ZhHh+bTrn6c/4KUovitG6Xo6am4aVGWWnlezN4m553YjNaVwvlmgsTiNtbf0Ems/lAK+HTC+HnF98Z/d6yOW/pAd9IeIkuTLU/mWZd2FeD2nxOv/HbO9t9viyfJ3/nf3r/JMB7T/pIf5grD/dIGsttOv8gQNDa4AGrQEatAZo0BqgQWuABq0BGrQGaNAaoEFrgAatARq0BmjQGqBBa4AGrQEatAZo0BqgQWuABq0BGrQGaNAaoEFrgAatARq0BmjQGqBBa4AGrQEatAZo0BqgQWuABq0BGrQGaNAaoEFrARj3bLztOHZEa/6L+7r3kEd9tOa92bvL05oHaM13v0WdLq35gNY8F/dfDe7mrY17Jw/TP5/858J0frY9L6ygNd8NJ5NiayZxbnteeInWAnCX24ectnYyiH810+zgFloLQLG1zud0Ewe30FoAiq0dfaM1J9FaAGjNC7QWAFrzAq0FgNa8QGsBoDUv0FoAiufXaM1RtBYAWvMCrQEatAZo0BqgQWuABq0BGrQGaNAaoEFrgAatARq0BmjQGqBBa4AGrQEatAZo0BqgQWuABq0BGrQGaNAaoEFrgAatARq0BmjQGqBBa4AGrQEatAZo0BqgQWuABq0BGrQGaNAaoPFfTmFg93C9xPMAAAAASUVORK5CYII=)
Again we observe that as observations flow in, Thompson sampling
figures out that treatment one is dominant and starts to solely assign
it. Such that at the end of the 1000 observations regret is negligible.
Note that the regret of a 50:50 experiment would have been 0.25.
Simulation study UCB
We now run this 100 times each for a sequence of values of the tuning
parameters alpha = c(0,0.5,1,1.5,2,2.5,3,4,5,10,1000)
. We
track the following quantities:
Average regret at \(i=10\),
\(i=100\) and \(i=1000\)
Probability to assign the optimal treatment at \(i=10\), \(i=100\) and \(i=1000\)
alpha = c(0,0.5,1,1.5,2,2.5,3,4,5,10,1000)
reps = 1000 # Reduce to 100 if you have no time
temp = matrix(NA,reps,6)
results = matrix(NA,length(alpha),6)
colnames(results) = c("regret10","regret100","regret1000","optimal10","optimal100","optimal1000")
for (a in 1:length(alpha)) {
for (j in 1:reps) {
# Draw sample
Y0 = rnorm(n,gamma0,sd0)
Y1 = rnorm(n,gamma1,sd1)
# Get kick-start
W = c(0,1,0,1)
Y = (1-W) * Y0[1:4] + W * Y1[1:4]
# Run UCB
for (i in 5:n) {
# Assign individual 4 to treatment one
Wi = ucb(Y,W,alpha[a])
W = c(W,Wi)
# and observe the potential outcome under treatment
Y = c(Y,(1-Wi) * Y0[i] + Wi * Y1[i])
}
regret = cumsum(gamma1 - (W * gamma1 + (1-W) * gamma0) ) / 1:n
temp[j,1] = regret[10]
temp[j,2] = regret[100]
temp[j,3] = regret[1000]
temp[j,4] = W[10]
temp[j,5] = W[100]
temp[j,6] = W[1000]
}
results[a,] = colMeans(temp)
}
as_tibble(cbind(alpha, results[,1:3])) %>% pivot_longer(!alpha,names_to = "individual", values_to = "regret") %>%
ggplot(aes(factor(alpha), regret, color=as.factor(individual))) +
geom_line(aes(group=factor(individual))) +
labs(x = "alpha", color = "Position") + theme_bw()
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAA2sAAAIcCAMAAABfO8ZvAAABNVBMVEUAAAAAADoAAGYAOjoAOmYAOpAAZrYAujgzMzM6AAA6ADo6AGY6OgA6Ojo6OmY6OpA6ZmY6ZpA6ZrY6kLY6kNtNTU1NTW5NTY5Nbm5NbqtNjshhnP9mAABmADpmOgBmOjpmOmZmZjpmZmZmZpBmkGZmkJBmkLZmkNtmtttmtv9uTU1uTW5uTY5ubm5uq+SOTU2OTW6OyP+QOgCQOjqQZgCQZjqQZmaQkDqQkGaQkLaQtpCQttuQ2/+rbk2r5P+2ZgC2kDq2kGa2tpC2tra2ttu229u22/+2/7a2/9u2///Ijk3I5KvI///bkDrbkGbbtmbbtpDb27bb29vb2//b/7bb/9vb///kq27k///r6+v4dm3/tmb/tpD/yI7/25D/27b/29v/5Kv//7b//8j//9v//+T////NvQZWAAAACXBIWXMAABJ0AAASdAHeZh94AAAgAElEQVR4nO2dC3vcxnWGl4pEms2lTqK1K65ip07bWL6Iltq6aW1KudRqxUpJ7SaiVqroUiTx/39CASywizvmHAxmzsz5vuexqV3ume8QOO8OLoOZRQJBkAstfCcAQUoE1iDIjcAaBLkRWIMgNwJrEORGYA2C3AisQZAbgTUIciMbrL0DQXJkoaLnkRXWxj+y5rfOD/URqcdUarpgjd866kimqdR0wRq/ddSRTFOp6YI1fuuoI5mmUtMFa/zWUUcyTaWmC9b4raOOZJpKTRes8VtHHck0lZouWOO3jjqSaSo1XbDGbx11JNNUarpgjd866kimqdR0wRq/ddSRTFOp6YI1fuuoI5mmUtMFa/zWUUcyTaWmC9b4raOOZJpKTRes8VtHHck0lZouWOO3jjqSaSo1XbDGbx11JNNUarpgjd866kimqdR0wRq/ddSRTFOp6YbM2ulyeefl5p+Xx8vl+8+S5OKT5e5NsKbTVGq6AbN2dufl9clR/s/L4/TnacrYmx1nmcCaRlOp6YbL2uXxgyR5k3Vm6Y/3HqV92qePkrOj2mfAmkZTD+m+ehU1axlaG+Aqb5zer30GrGk0dZ7uq1eR92t5l1ZlLT2GvDz+fHPelhRzQ64haGalfZqZgmdt25GdpceRF5+kLy8+e1a+h35No6nTdLM+zTAyeNbKfu1sWf6r0teBNY2mDtMtSYuctfr52ml2dWQjsKbc1Fm6O9IiZ616HTI5K65H5n3dF1vswJpGU0fpVkmLnLXsWkh5fy3v41Jdn2T84V62blM36b6qv4ybtWzcyFHG1/30ZC3Xg3z8SOV2NljTaOoi3VcN1JKD8ZiQWRsXWNNoOn+6LdLAGlhTaTp3uh2kJQeRH0OOC6xpNJ033S7SUtZW45Fgjd96fHUUh+mc6XaTlh5BgrXxj6COojOdL90e0sBaAtZ0ms6Vbi9pGWo4Xxv/COooOtOZ0u0lDaxlAmsaTWdJt79Ty6/3gzWwptF0hnSHSANrucCaRlPr6Q6TBtZygTWNppbTHSMtRy328ZDjAmsaTa2mO0oaWNsIrGk0tZiuAWn5SEiwBtZUmtpL14A0sFYIrGk0tZWuSadWDPAHa2BNpamddM1IK7s1sAbWNJraSNeUNLBWCqxpNJ2erjlp5SEkWANrGk2npksgDaxtBdY0mk5Ll0TaFjWwBtYUmr4a1rApjTSwthNYU2eawjIcOYHEtjYz+oC1BKypM81hcZguWNsKrGkyLbsld+kWE9WBtQSsaTIlLWNhyXTXrYE1sKbFlLiMhR1TsFYRWFNh+oq8jIUF06R2CAnWwJoC09bFQ7BGFVizGxmpacdlelfpllP4g7VMYC1u0+4bYo5ZW5lGgjV+62DNr2nfnWdH6da6NbAG1uI17R/jAdaoAmt2I6MyHRxN5Sbd7YJrYC0XWIvSdGwAMdsTrPEF1iI0HR0g7Ja1lXEkWOO3DtY8mBoMxXeSbqNbA2tgLS5Ts4dewBpVYM1uZPimpk+XuUh3txQ9WNsIrMVjav4cJ1ijCqzZjQzalPTEtIN0W6iBNbAWhSlxagKwRhVYsxvpxZQ7j0fVlBwM1qgCa3YjnZvmiK0rL2gz6KzLOLLx/NtohxpYKwXWPJm+Gp77w2TyqnXC6NIGPG2Gtrs1sAbWfJhW6TEK7SRvzSTNwTbq6NbAGlhzbdrsphimE87x2J60ULDWIbDm0rSLjxi3UQU1sLYVWHNm2tMPxbiNwFqXwJoT04EDvgi3USdqYA2szW46cmIV4TYCa50Ca7OaGlzAiHAbgbVOgbX5TOd+vkXqNqqiBtZ2emcNzaGsQ/Odgy8dVF+sKJGRszb+ETkdxcyRtkxpd76i20Y93Rr6NbBm2ZR8hzm6bQTWegTWLJqyhnLEto1qqIG1isCaJVP2kKnYthFY6xNYsxE6YWhidNsIrPUJrE0OrT6E5szUV+RoaC9qYA2sTQstejRso1JgrVdgbUJo9VEyZ6Z+I8dC66iBtarmZM3PE1auyr72x4G1QmCtX3OxtuHMx5PDTsq++S0C1jZqoAbWqpqDtV1/tk6cz4jhoDO1+sBn1KxVUQNrtll71XFoNWk+Npo2lwS5vJmY+l0U13vkcOhQtwbWbLLWLvH19jeElGim9QR2kZz5OEZNe5sEa7nA2pAssdZT2dVhguY5mZq2cmhFEnkbNB1sC6zlAmtDssCa4fP9NNrsXX4ndHD9pmNNgLVMg6iBtYmskZ7vn+9Jk9HL72a89ZgaxIK1TGBtUBNYYzzfP8+qRYbzLY53cF2hcz9cHRFrTdTAWl081kyPy9qh9lfj67se0/fpgcxboRaOPmcMBWvu5Ic1yvWG7o6CG9kl1q2uvj+hechrfswL1jpQA2t10VijXkfv3rwTTp2aDQ1e+xyLbUXX/1KzdkimdkOls1ZHDayZs8a5Qdx/qseNHG+EdqZXa2R3O1DggmYyIvtDx7o1sGbGmv2BGGPtGdxW5ka2GqoNKeM99wnWwNqoDP66mQYYTpoN2PZt5d1QadcTGYC1itSzNl8dMYmZac5u0YssSYnsDW2jBtYa8vysKONIcPLR5xyhgZk6Ya2BGljz/lw28QpHdEM4ImFtvFsDa95Z68Fn5rsFtkMDMwVrVEXCWidC3YNVLHraDQ3M1Hq6HaiBtaZEsNZBW8dwKdueNkMDMwVrVEXEWoul1sCwOTzthQZm6oK1JmpgTQxrDZ7Wvb+x6mkrNDBT2+madGtgTRBrNabWne/O4GknNDBTsEZVdKxVuFIzNDEG1rpQA2stCWOtMWuIgqGJYK0isMZvnTk2sYh0PLOkoOKVG9kVaoYaWBPIWjHLo/sZk8UUr+BIsMaXSNZme7ZgttDATK2m24kaWGtLKGtS6ihSU7BGFVizG6nHdH7W2qiBNbCm0dRmuqbdGlgDaxpNwRpVYM1upB5Ti+l2owbWOgTWNJqCNarAmt1IPab20jVHDayBNY2mYI0qsGY3Uo+ptXR7UANrXQJrGk3BGlVgzW6kHlOwRhVYsxupx9RWuhTUwBpY02gK1qgCa3Yj9ZhaSrcPNbDWKbCm0RSsUQXW7EbqMbWTbi9qYK1TYE2j6cysdaIWO2uny+Wdl5t/Xh4vl+8/q7+XgDWdpmCNqlHWzu68vD45yv95eZz+PE0hq7yXCaxpNLWSLvEQMm7WLo8fJMmbvDNL3rz3KEkuPn1UfS8TWNNoCtaoGmMtRasAbvdG8z2wptHURrr9qGlkLe++qqylx5DV997JtIYgjg56f7Nitxk8a/fL12fpcWTzPfRrGk0tpEvu1lT1a2fLB+2+DqxpNAVrVNHO106zqyOtcziwptF0eroDqKlkrXbN8WzzE9chYQrW6Bq9v3a6u5eW92eN9zKBNY2ms7LWg1rkrGVjRFKsrk/upydruR6U75UCaxpNJ6fL6NZiZ21cYE2jKVijCqzZjdRjOjXdIdTAWp/AmkbTOVnrQw2sgTWNphPTZXVrYA2saTQFa1SBNbuRekzBGlVgzW6kHtNp6Q6iBtZ6BdY0ms7IWi9qYA2saTSdlC6zWwNrYE2jKVijCqzZjdRjOiXdYdTAWr/AmkZTsEYVWLMbqcd0Qrps1MAaWNNoyo88AGtsgTWNptzIlLSRULDWL7Cm0ZQXmfdpYI0tsKbRlBF5UBw9gjW2wJpGU3Lk7jRtOHQANbAG1jSaEiOrF0TAGltgTaMpJfKgfukRrLEF1jSamke2rvGDNbbAmkZTw8iDjrtpYI0tsKbR1Ciy+7b1YOgQamANrGk0NYjsGyAC1tgCaxpNxyK7Dh5NQsHakMCaRtPhyMExj2CNLbCm0XQocmR08VDoIGpgDaxpNO2NHDh4NDAFa4MCaxpNeyJHQRsxBWuDAmsaTTsjTUgDaxME1jSatiPHDx4NTMHaoMCaRtNmpClow6bDqIE1sKbRtBZp3KWNmYK1YYE1jaaVSBJow6ZgbVhgTaPpNpJKGlibILCm0XQTSTt4HDcFa8MCaxpNs0gOaIOmI6hNZe35ItfNrzp/e/Xw1ovk/JdPN/+wL7BmN1KP6ZpLmn/WFot7Xb/NEXt+A6wFEqnHlE2aT9ZSkFL9adHPUvGROQTW7EaqMT2YJV03rCWPs5/ff5AeTT5JX13982Kx93Her/3lYdrp3d70a+Xv01e//9Fi79fj1iMCa3YjlZiOT17MM3XI2uvD7GBy7+uUpeKwss5a8/fdx50UgTW7kTpMDSYvZpmOoWbtGHI/I+jj7ATu1ovzw5+9SM4P9+vna7vfp//aT99PYyYKrNmNVGF6MM3TI2vFpZG0t8rgSvV47+vzwx/8/TfZv2us7X5/9TAj9O3d/XHvYYE1u5EaTI0mCmeZOmLtJ0+SlLX8mPB1it3j7L2fPmmyVv5+c/Z29XB/3HtYYM1uZPym5fXHIFnbXWSssJb89wcpbDeegjWwJsrUcFJ+nqlT1vazH48z1lJ9+0+Le73HkGBNZGTsprubajOkO4qaRdZ21z5eL37+Irn6bQHV8/zKY/3aCFgTGRm5qfECGCxTl6y1rukXUD3vuOYP1kRGRm1aGyoSOmvJee1e9s9ebKB6+8Fi/y85Xee7e9lgTWJkzKb1QVkhsuZXYM1uZMSmjfGP9tMdRw2sgTUFpsSFnTimYG1c76yh2HXgwGNlpZXIWRv/iOzvbIuRkZp2PD+Dfo0qsGY3Mk7TrkfVwBpVYM1uZIymjAULOaYGqIE1sBa1KWfBQo4pWDMQWIvZtG+qA7BGFVizGxmdKW9xUI4pWDMQWIvWdGACH7BGFVizGxmXKXshXoapCWpgDaxFajo4Kx1Yowqs2Y2MyZS/6DXHFKyZCKzFaDo21ypYowqs2Y2MxnR0VmOwRhVYsxsZi+n4BOKW0zVCLSbWrr781Wai89/9NWHxALAWnanBXP1gjao6a2/vbh4SP79LWUEArEVmarQsBlijqsLa27uLnSiL4oC1uEzNFqABa1RV+7X//ehvD/fe/SjTP/wPoQ2wFpWp4VpPmln7/kn2/7e/yOeWfGzaMzWOIT/8CWP1KbAWk6npsmp20zVDTQhr53+1gSyfx/V5NtndvklYx3XI7yh9WiawFo+p+QqG2llLT7n28h/3kuS10eWNJmvf/+NiceM/fkFa2A2sRWNKWCs0WtbOf/gv+eT+2RT/G6T+7YdPq2+eHy4Wt5PX+zlyBXcmi7M1WDs/3Pu7wxv/+ZC0sBtYi8WUsixvmKy96lb1I/lM/vlhYXp4+Pbu7fzqfO3N4hgy//F6AySDtceLe5uWcR1SnyltBewwWTNQvkJNiVD+83lGROXNDtZuGzTcvr9W/meeHFiLwpS41rzVdA1Rc8NahlCxUtvt5/ls4+kxZPVNK/0aWJsaGa4pEbXYWSsO7OqsFW9WWWOfr6Xnfvkx5Gvcy9ZmSkUtctZeF+uy5d1W+r/am1XW+Nch82sje785LBo1E1gL3pR2qjbRMwTWNstC7X29vTZSf/Pe9oPJY/b9tWwhnMXiJgU1sBa8KZ00u+maouaOtXxVtr3iNtpvCta2bz5e7Ce7W9qL/NW42vey/+/bb4g3s+dj7SATL5RvOikySFPWJo6WtbZeU86o+tU8X7slYIzWwVZ5KBu2EMvehylvA+tg7XW+sOi+lba6n6mhyQ5rdcKqodyuLcCy92DqfuuGxFp+nX/fTlONY8jn2TrdVE1hrY+weqjrb15FrHk4agiKNYtqPpf9YXry9+NUlPH+dNbGCWuEOj6jUMOal7PhVqgxajGx9vbDHxeagzUKYY1Qx1fKVLCW7wgR6apkjSeDv27CBUUfd4CiZ227O0SkC9aM5ej+msORDVGzVvvaE5GuStauvvyo1K++Mr5I4upetrsRe9Gy1jq8EJGuStbeZqNG9g43w5lvmp6yORs34mwkepSsdR7GS0jXHLWYWEueL249SZLvHy5uXz02vq3gbowWDTYJdSTEtPd0WUK6Olm7ergZdJw9K2r+vKjD8ZCkKyQS6kiA6eBlKQnp6mStHDdCe4bN6dhjJzNixMPa2OVfCenqZO2qmGgke37t/FAkawTYJNSRT1OT+ywS0tXJWnq+tvfxd9/97jA9X3so8Hwtl4NZ1cJnzfR+poB0CahFxVryx/wi5N7H6WGkvOuQhUxP2gTUkRdTyrgBAemqZS1Jvv3DH76hteH+WdG5Z5wPlzXeCLhpnhNDFbMWxLzHoRwfOTXljIMTsI3UshbKvMfzrloUHmvc8aYCtlGArHWunTG6hkaw8x4blJaAOqLoYIrcp2trG1FQE8Ja59oZ42toBDzvcRDXs411cBBYZ6qdtcbaGQZz14U8F+sYbP7ryFwHwR24Rssab+0MgzlZQ2ZtDDb/dWSq/BgQrFkxHa5Gg0Nw3toZBnONhz3vsfzxRyaa/NhmyKyRUHPTr7HWzjBYQyPweY8HLwp4ryMzlX8CWLNi6mntDHK/FuC8xwOwea8jE+2+LcCaFVNPa2eQz9cySZr32CRU9vMiY6pkD9asmHpaO4N8HbJr3uPT5fLOy/LF5ReP0v9ffLKsvul5Pv/BmSVn8rQUWjsEBmtWTH2tnTG+hsbovMdnd15enxyVr07fy1h7s+Msk++1M/pO2rzX0ZhsTf0RMGs01Nyxxlg7Y3QNjbF5jy+PH6Rsvf+seLHMWTs7qn3GN2t9XZvvOhpRezJ1B6YSIoWz1tY8a2e05j2++PRRAVyqN0f5y+T0fi3KP2vdsPmuo0G1O2OwZsU0mLUzWvMe511aydoWvc+Xy6KreyfT2r8OfCdAVGj5zqWV9RaDWTujpYK1siPLWbv4JH158dmz8jMC+rXOkzbf39n96jzDRL9mxTScZ2qa6urXcu3ek8Fax3Gk7zrqle3Ty3BZI6IWN2u18zXZrLVKWChr9i+bgrWKwmWtdh2yYG3T131RUieGtSZsXNMpz4N5uR0I1ioKl7XktHZ/LWft+uRB7R6bGNYaXQbDtBj0vR6dV7FXY0PK5hjmAtYqCpi1bNzIUcZXfnlkcwx5eVwdNiKItXqvQYysdGfr4g1SAwamMw2VBmsVhczauCSxxhtg2DxqXG/fp5mPmM71CFCwrFFRA2uiWKsUtOHl947+a139Nc2+33SsJbBmxRSs8Vsnhxo/pNI/Kc668TGCe6/pjNOjgLWKwBq/dXpoycZA5MjcU83I6RMJzzrtF1irCKzxW+eEDk0qYDLFW0ekKW49puzICaZSI8EaX/JY29R2K9J4JsU+Spmhcy9jESprZNTAmsFft6Jv1lK8fdqamIp0g7rPk7fI0uxLfYC1itSztk5pY+LG3KcpFbsr99SricNnerRQY2uwZsUUrGX/49HG3qfZRMLM+beHPSmXVSx0prOGgjV3cnu+xunc+PuUP6xxfFBjb9P1UDfLewfKGuOrVwhrg2tn9K6h4fzaCBk3qXVkcGvO1VJoYrfRcGiwrA2undG/hoaP65A02uTWUWfnVgm1NuJkxlCwxtDQ2hkDc9f5ueZP6dxE11Ebt/XuV3OZ2gwFay1NWztjYE5Wb/fXjHGTXkd9w5ZnNbUWqo21VbeqH5m2dsbAXOM+72Wb0RZAHVVxWxfvzG5qJ9Qja5yr0k76tUlrZwysoeF33IhJ5xZGHdUefXO9nG4g26gRKpa1SWtnCO3Xco3iFkodFZ3bmnP4yDadHArWWpq2dobE87WKhmkLqI4y3Nbz3dSbIxSstTRt7Qxx1yGbGurcwqqjOWcFmiMUrLU0ce2M/jU0ZLCWDOAWZB0pMJ2a7lwD9ryvndG7hoYY1pK+Y8kQ60iDabSstTXL2hk82Xt+ratzC7GONJjqYG2+tTN4svqsaAu3EOtIg6kO1lyunWEi289l12kLsY40mCphzaIkslbv3EKsIw2mE9PlPTkM1sY/Qt8xW9wCrCMVpmCNKrGsJeWxZIB1pMIUrFElmbVN5xZgHakwBWtUyWYt8TAD17RIPaZgjSrxrCVrNmxqyj5A1uabVA2s8VsHazJNwRpVAbDG3S96yh6sVQTW+K2DNZmmYI2qEFibccdYj9RjKvXrE6zxW5/1opX1SD2mYI0qsGY3Uo8pWKMqDNZmG6hqPVKPKVijCqzZjdRjCtaoCoS1uSansB6px1TqLVOwxm+9CJ1nklzrkXpMwRpVYM1upB5TsEZVMKzNszCe9Ug9pmCNqnBYo8OmpuzBWkVgjd86WJNpOiHdWR+TAmv81neh1F2kpuzBWkVgjd86WJNpCtaoCok16j5SU/aBsbaa1RSs8VuvhtJgU1P2YK0isMZvHazJNGVHzjxdE1jjt14LJcGmpuyDYm01sylY47cO1mSaMiNXc5uCNX7r9VAKbGrKHqxVBNb4rTdCCbCpKfuAWJt/KuvIWVs71MqlGWRZDvZe5KyNf8Tid7Z5x6amiwmmX3OxHApY47cO1mSagjWqwmPNHDY1ZR8Ka+WeA2tsOWbNGDY1ZR8Ia9v9BtbYAmsaTcmRjpaKBWv81jtCDWFTU/ZgrSKwxm+9K9QMNjVlHwRrlX0G1tgCaxpNiZHVXQbW2HLPmhlsaso+ANZqOwyssQXWNJqCNaoCZc0INjVlL5+1+u4Ca2z5YM0ENjVlL561xs4Ca2yBNY2mYI2qYFkzgE1N2UtnrbmrwBpbYE2jqXlka0+BNbb8sDYOm5qyl81aez+BNbY8sTYKm5qyB2sVgTV+62BNpqlpZMdeAmts+WJtDDY1ZS+Zta59BNbYAmsaTc0iO3cRWGPLG2sjsKkpe7BWEVjjtz4YOgibmrKXy1r3/gFrbIE1jaYmkT27B6yx5ZG1QdjUlL1U1vp2DlhjyydrQ7CpKXuwVhFY47cO1mSajkf27hqwxpZX1gZgU1P2MlnztGPAGr91sCbTlL9fwBpfflnzdKii3hSsURUBa35Owe2HBmbKP48Ga3yBNY2m/HsxYI0v36x5uWVqPzQwU/4YA7DGF1jTaArWqIqCNR/D7uyHBmY6FOnzAQywxm/dKNT9cHL7oYGZDkR6fYgXrPFbB2syTcEaVZGw5v7xX/uhgZn2R/qddAms8VsHazJNeyM9TyYI1vitG4a6nkLGfmhgpn2RviekBmv81k1DHU+NZj80MFOwRhVYm+JpNTQw055I7wsIgTV+68ahbqeyth8amGl3pP+F8cAav3XzUKdLNNgPDcy0M1LA4spgjd86WJNpCtaoiok1p8t82Q8NzLQr0gw1sMYXWNNo2hFpiBpY40sMay6XZbYfGphpO9IUNbDGlxzW6rtbTdmDtYrAGr91sCbTtBVpjBpY40sQa7UdrqbsRbBmjhpY4wusaTRtRBJQA2t8SWKtus/VlD1Yqwis8Vunhu52upqyF8AaBTWwxhdYsxIamGktkoQaWOvT6XJ552X54vKLR633hLG22+9qyh6sVRQwa2d3Xl6fHJWvTt971HoPrNkJDcy0GklDDax16/L4QZK8ef9Z8WKZsVZ7LxHH2nbPqyl736wRUQNr3br4tIQr1Zuj/GXtvUQea+W+V1P2nlmjogbWupV3Xzuucsyq772TaS1MK98J6JKszR08a/eLl1XWyvfk9WvFN62aLsZvv0bu1tCvdWusX8skj7XN/ldT9l5Zo6MG1rrVODcL43wNrLmLZKAG1rrVuOa4w0zwdchMq0mmgZU9WKsoXNaS09q9tJy1xntgzU5oYKabSA5qYK1Pp8tlitX1SX4pZMNa8V4piaxlRaCm7P2xxkINrPElkrW0DNSUvTfWeKiBNb7AmpXQwEzBGlXxspas1JS9L9aYqIE1vsCaldDATNds1MAaX0JZS1YeikERa+ytC9b4ksramk1bYGUP1ioCa/zWJ569O74BpIc1PmpgjS/JrGW0uRwcq4W11YSzYbDGl2zWEkbnJrWOxJhOGikA1vgSzxqZNt/pSjedNgIOrPEVAGvEQ0n/6Uo2XU18igKs8RUEawmlcxORrlTTyTNMgDW+QmHNnDYh6Uo03W5CqemCNX7rdneM2aGkmHTFmdqY6Ras8RUSa4lR5yYpXUmm1U0nNV2wxm99hh0zSpusdMWYWlrdDqzxFRxro7RJS1eGqa0VksEaXwGyNnLiJi9d/6bN7SU1XbDGb32+HdNPm8h0/Zq2tpXUdMEav/U5d0wfbULT9WfasaGkpgvW+K3PW0fdh5Ji0/VkGtI2Amv81vGd7d20s/eXmi5Y47fu4lykWUyy03Vs2nOgLTVdsMZv3UnxNg4lpafr0rTvApLQdMGa1B1TUQhjItybhnetFqzxW3c41s/C6r+RsRbgPUiwxm/dZfGWh5KBpDu36eDoGnnpbgTW+K07Lt6V5GeO3ZqGOY4NrPFbd168qymzk0fE2sgAbWnplgJr/NY9FC9/Etd4WAv2WQiwxm/dR/GufUyZLGobhfuMH1jjt+6FNX7fFgVrJn+8oHRrAmv81j2x5n7KZDnbyOgPl5NuXWCN37o31ni0RcCa2V8tJt2GwBq/dY+scWgLnrXQ5xoDa/zWvbJGXyIidNaCn0MTrPFb98yau+nJJWwjwh8rId0ugTV+675ZI9IWNGtu/lCwxlfsrDn6uve/jUg9uP90uwXW+K1LYI1AW7isRbKWD1jjty6DNeNSDJa1WNaoA2v81qWwZkhboKy5vLkB1vjSwprRdz/blLPW8GTT7V17l55gjS89rBl8/fNMV/mDPGzafIxGk7pLwRq/dVGsjdclw3S1W6iTSdu0v9S1J1jjSxVrYzxQTXeHjmuD1i2Z1iKdP9AA1vhSxtowDyTT2jnauutNM034Sz0ct4I1vtSxZmcqtyZT/FtdUy7IcCPF7lKwxm9dJGuTpyjt6LzqQzgcjOFYTZpZReouBWv81oWyNmXqbaMFOyiHkpy/dHdBhiepuxSs8VuXyloPbQaHn8YLUc03MGybBFhzJ7A2JZK6xM1qqLNyNjCsmgRYcycrrK31arUifNT8syyDGbMISJGzNv4RqV+CFiL7rybWP9sjaz4AAAXsSURBVMUecTIhtKMtbiTf02Yo+rXxj0jdMVYiawXcfSQ4cdSyncEqXWmANXcCazYih5aTMr6eOO9glZ40wJo7gTUrkX3LSVm8cD9lsEp/GmDNncCapciOi+jEwVb8uwWjFz8nmM4QCdb4Amu56jeH6aMa57mBPpIGWHMnsGYxMqvr9eYfsz18SbqBPp4HWHMnsGY1crVasx+wtn0D3SgPsOZOYM1u5ISR8+amLYq6hndFPhVRr8Aav/XAWHNkWkepGWoOfHzbCKzxWwdr3ep6zLT4BaFrjW8bgTV+62CtT83pE+rvzWRqIRKs8QXWvJkWZK3LV/qWsGoJrPFbB2uDyvFab//lxnRaJFjjC6x5NS2mlvQx253zULA2/hGpO8Z6pBfTCVMmx7eNwBq/dbAm01RqumCN3zrqSKap1HTBGr911JFMU6npgjV+66gjmaZS0wVr/NZRRzJNpaYL1vito45kmkpNF6zxW0cdyTSVmi5Y47eOOpJpKjVdsMZvHXUk01RqumCN3zrqSKap1HTBGr911JFMU6npgjV+66gjmaZS0wVr/NZRRzJNpaYL1vito45kmkpNF6zxW0cdyTSVmi5Y47eOOpJpKjVdsMZvHXUk01RqumCN3zrqSKap1HTBGr911JFMU6npgjV+66gjmaZS0wVr/NZRRzJNpaYbOWsQJEcWKnoe2WDNQBM2AD/UR6Qe08DSFSCwZjdSj2lg6QoQWLMbqcc0sHQFyBFrEKReYA2C3AisQZAbgTUIciOwBkFuBNYgyI2csHa6XN55OSHo4pMlo4XLLx5RPSsxLM/L4+Xy/Wf8GJbp9clyeZ8WstPFp/SNlHD3SLLdwISKaEQ0fwYkF6yd3Xl5fXI0IegNa6uevkcvo10Mx/PyOE34lBZYi2H9oadHaSNM2K5PGBsp4e6RpNzAlIqoRzR/hiQHrF0eP0h3DvHrvhZ0xtioaXdBLqNqDMfzTRZN7ClqMRzTPPaMWftnf8Pr1ziJJtsNTKiIRkTzJysLX3LAWl4M+cbhBp0yvrXfHNEPj6oxHM9cnKOyMoZtymTt4rP/4rHGTLTYwISKaEQ0f7Ky8CUHrOVfP9TtUg26PP6cfho0rei5ngn5GLIawzZlVt31yQPe+Rp/6+R+pIqoRTR/MjLwJ3es0b4Jq0EXn9zPvoKpu3YSa0zPtIdhnACVMUzT6xPmZYL0SJDHGnvr1Mgxq4haRPMnPQGPCqFfy0X/EpvEGtMzOVvSv2vrMaxva9apSwYL8zpkJlai6Ndm1fTztUxhsDbt2ifTlBt0tszFLVg+a6SKwPkaQZOvQ26+xMh3yyaxxvM8Y3QvlRiW6bSveF6/xtwjSRUz04qoReA65IhOOfdCKkHpKTznjs4k1lieU3tSlilz63TYu/Dc+FEqoh7R/BmSXI0boW+WTdD1yf3N0Ar6jmVXPtuTc1BWxvD/0GzcCHsIBfN8jbdHdn6EimhENH8GJIyHhCA3AmsQ5EZgDYLcCKxBkBuBNQhyI7AGQW4E1iDIjcAaBLkRWPOqq4e3XvS9giITWPMqsKZIYM2rwJoigTUv+tOPFou9X2/oenv39uvDxc2v81d//iB/f/cJKBqBNR96vsh1r2Dt3cP0xd7X6aubH2ze330CikZgzYOuHt54kiTnh/sFa4v9F1e/XWSvFulB5B/zf5WfgKIRWPOj737/5eGiZC0/S3t842lK2NMkeXt3v/IJKBqBNR86P8yPEEvW9rP3nu99vbk2cvVwv/IJKBqBNQ9Kjxrf/dd///PdftZ2n4CiEVjzoPPDe0n1fG1DWHYMWbK2+wQUjcCaB6UMvUi+/2Bx60VxbeTn5bWRHWvlJ3znClkTWPOgq4ebK/olaz/ITs5uPK2wtvuE71whawJrPpT2WIubXz3eHDWm52uvDxc/fZJUWNt9wneqkDWBNe96iysgOgTWvAusKRFY8y6wpkRgzbvAmhKBNQhyI7AGQW4E1iDIjcAaBLkRWIMgNwJrEORGYA2C3AisQZAbgTUIcqP/BySm4g7KhqA6AAAAAElFTkSuQmCC)
as_tibble(cbind(alpha, results[,4:6])) %>% pivot_longer(!alpha,names_to = "individual", values_to = "optimal") %>%
ggplot(aes(factor(alpha), optimal, color=as.factor(individual))) +
geom_line(aes(group=factor(individual))) +
labs(x = "alpha", y = "Prob of optimal choice", color = "Position") + theme_bw()
![](data:image/png;base64,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)
We observe that a value of \(\alpha =
3\) minimizes regret in the long run and maximizes the
probability to assign the best treatment to individual 1000. Very low
values of \(\alpha\) result in too
little exploration. This means the bandit might commit early on to the
wrong treatment. On the other hand, very high values of \(\alpha\) explore too much. \(\alpha = 1000\) is a very extreme case,
which is nearly equivalent to running a 50:50 experiment. The bandit
ignores basically any knowledge about the better treatment.
Simulation study Thompson sampling
Use the Thompson function in the same loop as before:
alpha = c(0,0.5,1,1.5,2,2.5,3,4,5,10,1000)
reps = 1000 # Reduce to 100 if you have no time
temp = matrix(NA,reps,6)
results = matrix(NA,length(alpha),6)
colnames(results) = c("regret10","regret100","regret1000","optimal10","optimal100","optimal1000")
for (a in 1:length(alpha)) {
for (j in 1:reps) {
# Draw sample
Y0 = rnorm(n,gamma0,sd0)
Y1 = rnorm(n,gamma1,sd1)
# Get kick-start
W = c(0,1,0,1)
Y = (1-W) * Y0[1:4] + W * Y1[1:4]
# Run UCB
for (i in 5:n) {
# Assign individual 4 to treatment one
Wi = thompson(Y,W,alpha[a])
W = c(W,Wi)
# and observe the potential outcome under treatment
Y = c(Y,(1-Wi) * Y0[i] + Wi * Y1[i])
}
regret = cumsum(gamma1 - (W * gamma1 + (1-W) * gamma0) ) / 1:n
temp[j,1] = regret[10]
temp[j,2] = regret[100]
temp[j,3] = regret[1000]
temp[j,4] = W[10]
temp[j,5] = W[100]
temp[j,6] = W[1000]
}
results[a,] = colMeans(temp)
}
as_tibble(cbind(alpha, results[,1:3])) %>% pivot_longer(!alpha,names_to = "individual", values_to = "regret") %>%
ggplot(aes(factor(alpha), regret, color=as.factor(individual))) +
geom_line(aes(group=factor(individual))) +
labs(x = "alpha", color = "Position") + theme_bw()
![](data:image/png;base64,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)
as_tibble(cbind(alpha, results[,4:6])) %>% pivot_longer(!alpha,names_to = "individual", values_to = "optimal") %>%
ggplot(aes(factor(alpha), optimal, color=as.factor(individual))) +
geom_line(aes(group=factor(individual))) +
labs(x = "alpha", y = "Prob of optimal choice", color = "Position") + theme_bw()
![](data:image/png;base64,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)
We see the same exploration vs. exploitation trade-off as for UCB.
However, the optimal \(\alpha = 1\) and
Thompson sampling manages to realize a smaller average regret of around
0.05, while UCB had only 0.07, at least with this very coarse grid.
Potential extensions
- Change the mean and/or variance of the treatment and control group.
First think about what you expect and then test your intuition by
running the modified notebook.
---
title: "Causal ML: Multi-armed bandit"
subtitle: "Simulation notebook"
author: "Michael Knaus"
date: "`r format(Sys.time(), '%m/%y')`"
output:
  html_notebook:
    toc: true
    toc_float: true
    code_folding: show
---

<br>

Goals:

-   Handcode UCB and Thompson sampling in stylized DGP

<br>

# Bandits

## One step

To illustrate bandits, we consider a very simple DGP:

-   $Y(0) \sim N(5,2^2)$

-   $Y(1) \sim N(5.5,2^2)$

This means that the optimal policy rule assigns everybody to treatment one. Let's see how UCB and Thompson sampling figure this out on-the-fly. For a stream of 1000 observations.

```{r, warning=F,message=F}
if (!require("tidyverse")) install.packages("tidyverse", dependencies = TRUE); library(tidyverse)
if (!require("ggridges")) install.packages("ggridges", dependencies = TRUE); library(ggridges)

set.seed(1234)

# Define parameters
gamma0 = 5
gamma1 = 5.5
sd0 = 2
sd1 = 2
n = 1000

# Draw the potential outcomes of all individuals
Y0 = rnorm(n,gamma0,sd0)
Y1 = rnorm(n,gamma1,sd1)

hist(Y0)
hist(Y1)
```

As a kick-start, we assign the first four individuals "manually" to control/treatment/control/treatment:

```{r}
# Assign individual 1 to treatment zero
W = 0
# and observe the potential outcome under non-treatment
Y = Y0[1]
# Assign individual 2 to treatment one
W = c(W,1)
# and observe the potential outcome under treatment
Y = c(Y,Y1[2])
# Assign individual 3 to treatment zero
W = c(W,0)
# and observe the potential outcome under treatment
Y = c(Y,Y0[3])
# Assign individual 4 to treatment one
W = c(W,1)
# and observe the potential outcome under treatment
Y = c(Y,Y1[4])

# Print the realizes values
W
Y
```

Now calculate the means in each treatment arm:

```{r}
gamma = c(mean(Y[W==0]), mean(Y[W==1]))
gamma
```

And the standard errors of the means:

```{r}
se = c(sd(Y[W==0]) / sqrt(sum(W==0)),
       sd(Y[W==1]) / sqrt(sum(W==1)))
se
```

### Upper Confidence Bound (UCB)

For $i=5$, we implement now explicitly UCB $$W_{i+1} = argmax_w (\hat{\gamma}_{i,w} + \alpha \hat{\sigma}_{i,w})$$ with `alpha = 2`:

```{r}
# Calculate upper confidence bound
alpha = 2
upper_ci = gamma + alpha * se
upper_ci
# Choose the treatment with the highest bound
which.max(upper_ci)-1
```

Let's visualize the decision.

```{r}
plot_ucb = function(gamma, se, alpha, unit = FALSE) {
  # Calculate UCB and find the arm with the highest UCB
  ucb <- gamma + alpha * se
  highest_ucb_arm <- which.max(ucb)

  # Data frame for plotting
  data <- data.frame(
    Arm = factor(1:(length(gamma))),
    Estimate = gamma,
    UCB = ucb,
    Highlight = ifelse(1:length(gamma) == highest_ucb_arm, "forestgreen", "orange")
  )

  # Plot
  g = ggplot(data, aes(x = Arm)) +
    geom_errorbar(aes(ymin = Estimate, ymax = UCB, color = Highlight), width = 0.01, linewidth = 1) +
    geom_segment(aes(x = as.numeric(Arm)-0.05, y = UCB, xend = as.numeric(Arm)+0.05, yend = UCB, color = Highlight), linewidth = 1) +
    geom_point(aes(y = Estimate), size = 4, color = "black") +
    scale_color_identity() + geom_hline(yintercept = 0) +
    labs(y = "Estimate / Upper Confidence Bound", x = "Arm",
      title = "UCB Sampling for Multi-Armed Bandits") +
    theme_minimal() + ylim(0,10) + theme(plot.title =  element_text(hjust = 0.5))
  if (is.numeric(unit)) g = g + labs(title = paste("UCB Sampling of treatment for unit",unit+1))
  print(g)
}
plot_ucb(gamma,se,2)
```

### Thompson sampling

Thompson sampling draws a value for each arm from an arm specific normal distribution $\tilde{\gamma}_{i,w} \sim N(\hat{\gamma}_{i,w},\alpha^2\hat{\sigma}^2_{i,w})$ and sets $$W_{i+1} = argmax_w \tilde{\gamma}_{i,w} $$

```{r}
# Thompson
draws = c(rnorm(1,gamma[1],alpha*se[1]),
          rnorm(1,gamma[2],alpha*se[2]))
draws
which.max(draws)-1
```

Let's visualize the decision.

```{r}
plot_ts = function(gamma, se, alpha, gamma_tilde=draws, unit = FALSE) {
  # This is probably the most complicated and unelegant plot I ever created, but it makes the point
  # Let me know if you have an alternative to plot it more elegantly

  highest_arm <- which.max(gamma_tilde)

  # Data frame for plotting
  data <- data.frame(
    Arm = 0:1,
    Estimate = gamma,
    Draw = gamma_tilde,
    se = se * alpha
  )

  plot_data <- data %>%
    group_by(Arm) %>%
    slice(rep(1:n(), each = 10000)) %>%
    mutate(
      y = rnorm(n(), mean = Estimate, sd = se),
    ) %>%
    mutate(
      dens = dnorm(y,mean = Estimate, sd = se),
    ) %>%
    ungroup()

  # Plot
  g = ggplot(plot_data, aes(y=Arm)) + 
    geom_point(data=data, aes(y=Arm, x = Draw,color=Draw), size = 5, shape = 4, stroke=1) +
    scale_color_gradient(low = "orange", high = "forestgreen") +
    geom_point(data=data, aes(x=Estimate), size = 3, color = "black") +
    geom_density_ridges(data = plot_data[plot_data$Arm=="0",], aes(y=Arm,x = y, height = dens),
                        alpha=0.7, position = position_nudge(y = 0), stat = "identity") +
    geom_density_ridges(data = plot_data[plot_data$Arm=="1",], aes(y=Arm,x = y, height = dens),
                      alpha=0.7, position = position_nudge(y = 0),  stat = "identity") +
    labs(y = "Arm / Normal distribution", x = "Estimate",
         title = "Thompson Sampling") +
    theme_minimal() + scale_y_continuous(breaks = c(0, 1), labels = c("0", "1")) +
    # # xlim(-10,20) +
    geom_vline(xintercept = 0) +
    theme(legend.position="none", plot.title = element_text(hjust = 0.5)) +
    coord_flip(ylim = c(0, 3),xlim = c(-10, 20))

    if (is.numeric(unit)) g = g + labs(title = paste("Thompson Sampling for unit",unit+1))
  print(g)
}
plot_ts(gamma,se,2,draws)
```



Note that with this very seed, UCB and Thompson sampling do not take the same choice $\Rightarrow$ their treatment sequences differ. Note that the decision of UCB is deterministic, while it is random for Thompson sampling.

<br>

## Dynamic treatment assignment (UCB)

First, proceed with UCB and define a function that takes

-   observed outcomes

-   assigned treatments

-   and the tuning parameter $\alpha$ as arguments

and returns a treatment assignment for the next individual:

```{r}
ucb = function(Y, W, alpha, plot = FALSE){
  gamma = c(mean(Y[W==0]), mean(Y[W==1]))
  se = c(sd(Y[W==0]) / sqrt(sum(W==0)), sd(Y[W==1]) / sqrt(sum(W==1)))
  assign = which.max(gamma + alpha * se)
  if (isTRUE(plot)) plot_ucb(gamma,se,2, unit = length(Y))
  return(assign-1)
}
```

Now we run UCB until $i=1000$ and plot the evolution:

```{r}
# Store to use it for Thompson below
Wstart = W
Ystart = Y

plot_grid = c(5:10,20,50,100,200,500,1000)
for (i in 5:1000) {
  # Assign unit to treatment (and sometimes plot)
  if (i %in% plot_grid) Wi = ucb(Y,W,alpha,plot = TRUE)
  else Wi = ucb(Y,W,alpha,plot = FALSE)
  W = c(W,Wi)
  # and observe the potential outcome under treatment
  Y = c(Y,(1-Wi) * Y0[i] + Wi * Y1[i])
}
```

additionally plot the share being assigned to the optimal policy

```{r}
share = cumsum(W) / 1:n
plot(1:n,share)
```

and the regret

```{r}
regret = cumsum(gamma1 - (W * gamma1 + (1-W) * gamma0) )  / 1:n
plot(1:n,regret)
```

We observe that as observations flow in, UCB figures out that treatment one is dominant and starts to solely assign it. Such that at the end of the 1000 observations regret is negligible. Note that the regret of a 50:50 experiment would have been 0.25.

<br>

## Dynamic treatment assignment (Thompson)

Write a function similar to `ucb` above:

```{r}
thompson = function(Y, W, alpha, plot = FALSE){
  gamma = c(mean(Y[W==0]), mean(Y[W==1]))
  se = c(sd(Y[W==0]) / sqrt(sum(W==0)), sd(Y[W==1]) / sqrt(sum(W==1)))
  draws = c(rnorm(1, gamma[1], alpha * se[1]),
          rnorm(1, gamma[2], alpha * se[2]))
  assign = which.max(draws)
  if (isTRUE(plot)) plot_ts(gamma,se,2,draws,unit = length(Y))
  return(assign-1)
}
```

Now we run Thompson until $i=1000$ and plot the evolution:

```{r}
# Restore starting points
W = Wstart
Y = Ystart

plot_grid = c(5:10,20,50,100,200,500,1000)
for (i in 5:1000) {
  # Assign unit to treatment (and sometimes plot)
  if (i %in% plot_grid) Wi = thompson(Y,W,alpha,plot = TRUE)
  else Wi = thompson(Y,W,alpha,plot = FALSE)
  W = c(W,Wi)
  # and observe the potential outcome under treatment
  Y = c(Y,(1-Wi) * Y0[i] + Wi * Y1[i])
}
```

additionally plot the share being assigned to the optimal policy

```{r}
share = cumsum(W) / 1:n
plot(1:n,share)
```

and the regret

```{r}
regret = cumsum(gamma1 - (W * gamma1 + (1-W) * gamma0) )  / 1:n
plot(1:n,regret)
```

Again we observe that as observations flow in, Thompson sampling figures out that treatment one is dominant and starts to solely assign it. Such that at the end of the 1000 observations regret is negligible. Note that the regret of a 50:50 experiment would have been 0.25.

<br>

## Simulation study UCB

We now run this 100 times each for a sequence of values of the tuning parameters `alpha = c(0,0.5,1,1.5,2,2.5,3,4,5,10,1000)`. We track the following quantities:

-   Average regret at $i=10$, $i=100$ and $i=1000$

-   Probability to assign the optimal treatment at $i=10$, $i=100$ and $i=1000$

```{r}
alpha = c(0,0.5,1,1.5,2,2.5,3,4,5,10,1000)

reps = 1000 # Reduce to 100 if you have no time

temp = matrix(NA,reps,6)
results = matrix(NA,length(alpha),6)
colnames(results) = c("regret10","regret100","regret1000","optimal10","optimal100","optimal1000")

for (a in 1:length(alpha)) {
  for (j in 1:reps) {
    # Draw sample
    Y0 = rnorm(n,gamma0,sd0)
    Y1 = rnorm(n,gamma1,sd1)

    # Get kick-start
    W = c(0,1,0,1)
    Y = (1-W) * Y0[1:4] + W * Y1[1:4]

    # Run UCB
    for (i in 5:n) {
      # Assign individual 4 to treatment one
      Wi = ucb(Y,W,alpha[a])
      W = c(W,Wi)
      # and observe the potential outcome under treatment
      Y = c(Y,(1-Wi) * Y0[i] + Wi * Y1[i])
    }
    regret = cumsum(gamma1 - (W * gamma1 + (1-W) * gamma0) )  / 1:n
    temp[j,1] = regret[10]
    temp[j,2] = regret[100]
    temp[j,3] = regret[1000]
    temp[j,4] = W[10]
    temp[j,5] = W[100]
    temp[j,6] = W[1000]
  }
 results[a,] = colMeans(temp)
}

as_tibble(cbind(alpha, results[,1:3])) %>% pivot_longer(!alpha,names_to = "individual", values_to = "regret") %>%
  ggplot(aes(factor(alpha), regret, color=as.factor(individual))) +
  geom_line(aes(group=factor(individual))) +
  labs(x = "alpha", color = "Position") + theme_bw()

as_tibble(cbind(alpha, results[,4:6])) %>% pivot_longer(!alpha,names_to = "individual", values_to = "optimal") %>%
  ggplot(aes(factor(alpha), optimal, color=as.factor(individual))) +
  geom_line(aes(group=factor(individual))) +
  labs(x = "alpha", y = "Prob of optimal choice", color = "Position") + theme_bw()
```

We observe that a value of $\alpha = 3$ minimizes regret in the long run and maximizes the probability to assign the best treatment to individual 1000. Very low values of $\alpha$ result in too little exploration. This means the bandit might commit early on to the wrong treatment. On the other hand, very high values of $\alpha$ explore too much. $\alpha = 1000$ is a very extreme case, which is nearly equivalent to running a 50:50 experiment. The bandit ignores basically any knowledge about the better treatment.

<br>

## Simulation study Thompson sampling

Use the Thompson function in the same loop as before:

```{r}
alpha = c(0,0.5,1,1.5,2,2.5,3,4,5,10,1000)

reps = 1000 # Reduce to 100 if you have no time

temp = matrix(NA,reps,6)
results = matrix(NA,length(alpha),6)
colnames(results) = c("regret10","regret100","regret1000","optimal10","optimal100","optimal1000")

for (a in 1:length(alpha)) {
  for (j in 1:reps) {
    # Draw sample
    Y0 = rnorm(n,gamma0,sd0)
    Y1 = rnorm(n,gamma1,sd1)

    # Get kick-start
    W = c(0,1,0,1)
    Y = (1-W) * Y0[1:4] + W * Y1[1:4]

    # Run UCB
    for (i in 5:n) {
      # Assign individual 4 to treatment one
      Wi = thompson(Y,W,alpha[a])
      W = c(W,Wi)
      # and observe the potential outcome under treatment
      Y = c(Y,(1-Wi) * Y0[i] + Wi * Y1[i])
    }
    regret = cumsum(gamma1 - (W * gamma1 + (1-W) * gamma0) )  / 1:n
    temp[j,1] = regret[10]
    temp[j,2] = regret[100]
    temp[j,3] = regret[1000]
    temp[j,4] = W[10]
    temp[j,5] = W[100]
    temp[j,6] = W[1000]
  }
 results[a,] = colMeans(temp)
}

as_tibble(cbind(alpha, results[,1:3])) %>% pivot_longer(!alpha,names_to = "individual", values_to = "regret") %>%
  ggplot(aes(factor(alpha), regret, color=as.factor(individual))) +
  geom_line(aes(group=factor(individual))) +
  labs(x = "alpha", color = "Position") + theme_bw()

as_tibble(cbind(alpha, results[,4:6])) %>% pivot_longer(!alpha,names_to = "individual", values_to = "optimal") %>%
  ggplot(aes(factor(alpha), optimal, color=as.factor(individual))) +
  geom_line(aes(group=factor(individual))) +
  labs(x = "alpha", y = "Prob of optimal choice", color = "Position") + theme_bw()
```

We see the same exploration vs. exploitation trade-off as for UCB. However, the optimal $\alpha = 1$ and Thompson sampling manages to realize a smaller average regret of around 0.05, while UCB had only 0.07, at least with this very coarse grid.

<br>

## Potential extensions

-   Change the mean and/or variance of the treatment and control group. First think about what you expect and then test your intuition by running the modified notebook.
