The Slot Bandit
The One Armed Bandit
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Play for real | Release Date | July 24, 2019 |
| Default bet size | €1.00 |
| Default Coin range | €0.01 - €5.00 |
| Paylines | 20 |
| Default bet range | €0.10 - €40 |
| Max RTP | 96.30% |
| Volatility | 91 VERY HIGH |
| Maximum Win | 11800 x / €472,000 |
Table Of Contents
Every slot spinner we know loves a high noon session. And with the release of Yggdrasil’s new The One Armed Bandit slot – we’ve uncovered the perfect duel to spin today when the boss isn’t looking! This Western themed fruit machine play like a classic One Armed Bandit fruit machine with spins from 0.10 to 40.00. Fancy joining the good guys in this dangerous town to spin for potential jackpots up to €472 000 with the max bet? Then we recommend you debrief with our full The One Armed Bandit slot review.
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SLOT PLOT
You know the story….
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RTP, STRATEGY & MOBILE
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WIDE RANGING BETS + LAWFUL WINS
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PLAY FOR FREE
You can sample some high noon gun-slinging by playing our The One Armed Bandit free play demo – before having a real duel at our top-rated slot sites.
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The One Armed Bandit slots has a range of wild bonuses that you can aim to hit during slot sessions:
SCATTER SYMBOL
The first four Scatter symbols that land on the reels aware you one Respin each. Landing scatters transform into Wilds and give you a bonus spin with enhanced chances to create winning combinations. If you land a fifth Scatter on a single spin or Respin round, you’ll receive an extra 10 Free Spins.
X5 WILD SYMBOL
Wilds can really power-up your sessions at One Armed Bandit slot sites, boosting wins with a non-stacked x5 multiplier. Wilds only appear on marked reels in Respins, but on all reels in Free Spins.
FREE SPIN MODE
Landing a Scatter symbol on each reel in a single spin or Respin round (in the base game) awards you 10 Free Spins. All Free Spins start with a x5 multiplier and have a maximum of x20 multiplier (plus any extra Wild multipliers).
BONUS SYMBOL
Each Bonus symbol awards an extra Free Spin. Landing two Bonus symbols in Free Spins boosts your win by x5. Landing an additional two Bonus symbols awards an extra x10 multiplier on all line wins.
FAQs
- Where Can I Play One Armed Bandits Slot?
The One Armed Bandit slot is produced by Yggdrasil and can be played at our top-rated recommended casino. Other new releases to spin include Wild Mantra slot and Pirates Smugglers Paradise slot.
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There are no promises when you spin at The One Armed Bandit slot sites. However, you can potentially win up to 11800x you stake if you’re very lucky and hit the maximum win; a whopping €472 000 if using the max 40.00 bet.
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The one Armed Bandit is a very volatile game that combines plenty of risk and rewards. This means that random sessions can blow hot or cold! We recommend either chancing your arm when you’re feeling lucky and or play regular sessions with a good volume of spins (e.g. 100 spins). The big wins tend to come virtue of the free spins, so high-volume sessions should boost the chances of activating the bonus.
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The multi-armed bandit problem is a class example to demonstrate the exploration versus exploitation dilemma. This post introduces the bandit problem and how to solve it using different exploration strategies.
- What is Multi-Armed Bandit?
- Upper Confidence Bounds
The algorithms are implemented for Bernoulli bandit in lilianweng/multi-armed-bandit.
Exploitation vs Exploration
The exploration vs exploitation dilemma exists in many aspects of our life. Say, your favorite restaurant is right around the corner. If you go there every day, you would be confident of what you will get, but miss the chances of discovering an even better option. If you try new places all the time, very likely you are gonna have to eat unpleasant food from time to time. Similarly, online advisors try to balance between the known most attractive ads and the new ads that might be even more successful.
Fig. 1. A real-life example of the exploration vs exploitation dilemma: where to eat? (Image source: UC Berkeley AI course slide, lecture 11.)
If we have learned all the information about the environment, we are able to find the best strategy by even just simulating brute-force, let alone many other smart approaches. The dilemma comes from the incomplete information: we need to gather enough information to make best overall decisions while keeping the risk under control. With exploitation, we take advantage of the best option we know. With exploration, we take some risk to collect information about unknown options. The best long-term strategy may involve short-term sacrifices. For example, one exploration trial could be a total failure, but it warns us of not taking that action too often in the future.
What is Multi-Armed Bandit?
The multi-armed bandit problem is a classic problem that well demonstrates the exploration vs exploitation dilemma. Imagine you are in a casino facing multiple slot machines and each is configured with an unknown probability of how likely you can get a reward at one play. The question is: What is the best strategy to achieve highest long-term rewards?
In this post, we will only discuss the setting of having an infinite number of trials. The restriction on a finite number of trials introduces a new type of exploration problem. For instance, if the number of trials is smaller than the number of slot machines, we cannot even try every machine to estimate the reward probability (!) and hence we have to behave smartly w.r.t. a limited set of knowledge and resources (i.e. time).
Fig. 2. An illustration of how a Bernoulli multi-armed bandit works. The reward probabilities are unknown to the player.
A naive approach can be that you continue to playing with one machine for many many rounds so as to eventually estimate the “true” reward probability according to the law of large numbers. However, this is quite wasteful and surely does not guarantee the best long-term reward.
Definition
Now let’s give it a scientific definition.
A Bernoulli multi-armed bandit can be described as a tuple of (langle mathcal{A}, mathcal{R} rangle), where:
- We have (K) machines with reward probabilities, ({ theta_1, dots, theta_K }).
- At each time step t, we take an action a on one slot machine and receive a reward r.
- (mathcal{A}) is a set of actions, each referring to the interaction with one slot machine. The value of action a is the expected reward, (Q(a) = mathbb{E} [r vert a] = theta). If action (a_t) at the time step t is on the i-th machine, then (Q(a_t) = theta_i).
- (mathcal{R}) is a reward function. In the case of Bernoulli bandit, we observe a reward r in a stochastic fashion. At the time step t, (r_t = mathcal{R}(a_t)) may return reward 1 with a probability (Q(a_t)) or 0 otherwise.
It is a simplified version of Markov decision process, as there is no state (mathcal{S}).
The goal is to maximize the cumulative reward (sum_{t=1}^T r_t).If we know the optimal action with the best reward, then the goal is same as to minimize the potential regret or loss by not picking the optimal action.
The optimal reward probability (theta^{*}) of the optimal action (a^{*}) is:
[theta^{*}=Q(a^{*})=max_{a in mathcal{A}} Q(a) = max_{1 leq i leq K} theta_i]Our loss function is the total regret we might have by not selecting the optimal action up to the time step T:
[mathcal{L}_T = mathbb{E} Big[ sum_{t=1}^T big( theta^{*} - Q(a_t) big) Big]]Bandit Strategies
Based on how we do exploration, there several ways to solve the multi-armed bandit.
- No exploration: the most naive approach and a bad one.
- Exploration at random
- Exploration smartly with preference to uncertainty
ε-Greedy Algorithm
The ε-greedy algorithm takes the best action most of the time, but does random exploration occasionally. The action value is estimated according to the past experience by averaging the rewards associated with the target action a that we have observed so far (up to the current time step t):
[hat{Q}_t(a) = frac{1}{N_t(a)} sum_{tau=1}^t r_tau mathbb{1}[a_tau = a]]where (mathbb{1}) is a binary indicator function and (N_t(a)) is how many times the action a has been selected so far, (N_t(a) = sum_{tau=1}^t mathbb{1}[a_tau = a]).
According to the ε-greedy algorithm, with a small probability (epsilon) we take a random action, but otherwise (which should be the most of the time, probability 1-(epsilon)) we pick the best action that we have learnt so far: (hat{a}^{*}_t = argmax_{a in mathcal{A}} hat{Q}_t(a)).
Check my toy implementation here.
Upper Confidence Bounds
Random exploration gives us an opportunity to try out options that we have not known much about. However, due to the randomness, it is possible we end up exploring a bad action which we have confirmed in the past (bad luck!). To avoid such inefficient exploration, one approach is to decrease the parameter ε in time and the other is to be optimistic about options with high uncertainty and thus to prefer actions for which we haven’t had a confident value estimation yet. Or in other words, we favor exploration of actions with a strong potential to have a optimal value.
The Upper Confidence Bounds (UCB) algorithm measures this potential by an upper confidence bound of the reward value, (hat{U}_t(a)), so that the true value is below with bound (Q(a) leq hat{Q}_t(a) + hat{U}_t(a)) with high probability. The upper bound (hat{U}_t(a)) is a function of (N_t(a)); a larger number of trials (N_t(a)) should give us a smaller bound (hat{U}_t(a)).
In UCB algorithm, we always select the greediest action to maximize the upper confidence bound:
[a^{UCB}_t = argmax_{a in mathcal{A}} hat{Q}_t(a) + hat{U}_t(a)]Now, the question is how to estimate the upper confidence bound.
Hoeffding’s Inequality
If we do not want to assign any prior knowledge on how the distribution looks like, we can get help from “Hoeffding’s Inequality” — a theorem applicable to any bounded distribution.
Let (X_1, dots, X_t) be i.i.d. (independent and identically distributed) random variables and they are all bounded by the interval [0, 1]. The sample mean is (overline{X}_t = frac{1}{t}sum_{tau=1}^t X_tau). Then for u > 0, we have:
The Bandit Slot Player
[mathbb{P} [ mathbb{E}[X] > overline{X}_t + u] leq e^{-2tu^2}]Given one target action a, let us consider:
- (r_t(a)) as the random variables,
- (Q(a)) as the true mean,
- (hat{Q}_t(a)) as the sample mean,
- And (u) as the upper confidence bound, (u = U_t(a))
Then we have,
[mathbb{P} [ Q(a) > hat{Q}_t(a) + U_t(a)] leq e^{-2t{U_t(a)}^2}]We want to pick a bound so that with high chances the true mean is blow the sample mean + the upper confidence bound. Thus (e^{-2t U_t(a)^2}) should be a small probability. Let’s say we are ok with a tiny threshold p:
[e^{-2t U_t(a)^2} = p text{ Thus, } U_t(a) = sqrt{frac{-log p}{2 N_t(a)}}]UCB1
One heuristic is to reduce the threshold p in time, as we want to make more confident bound estimation with more rewards observed. Set (p=t^{-4}) we get UCB1 algorithm:
The Slot Bandit Band
[U_t(a) = sqrt{frac{2 log t}{N_t(a)}} text{ and }a^{UCB1}_t = argmax_{a in mathcal{A}} Q(a) + sqrt{frac{2 log t}{N_t(a)}}]Bayesian UCB
In UCB or UCB1 algorithm, we do not assume any prior on the reward distribution and therefore we have to rely on the Hoeffding’s Inequality for a very generalize estimation. If we are able to know the distribution upfront, we would be able to make better bound estimation.
For example, if we expect the mean reward of every slot machine to be Gaussian as in Fig 2, we can set the upper bound as 95% confidence interval by setting (hat{U}_t(a)) to be twice the standard deviation.
Fig. 3. When the expected reward has a Gaussian distribution. (sigma(a_i)) is the standard deviation and (csigma(a_i)) is the upper confidence bound. The constant (c) is a adjustable hyperparameter. (Image source: UCL RL course lecture 9’s slides)
Check my toy implementation of UCB1 and Bayesian UCB with Beta prior on θ.
Thompson Sampling
Thompson sampling has a simple idea but it works great for solving the multi-armed bandit problem.
Fig. 4. Oops, I guess not this Thompson? (Credit goes to Ben Taborsky; he has a full theorem of how Thompson invented while pondering over who to pass the ball. Yes I stole his joke.)
At each time step, we want to select action a according to the probability that a is optimal:
[begin{aligned}pi(a ; vert ; h_t) &= mathbb{P} [ Q(a) > Q(a'), forall a' neq a ; vert ; h_t] &= mathbb{E}_{mathcal{R} vert h_t} [ mathbb{1}(a = argmax_{a in mathcal{A}} Q(a)) ]end{aligned}]where (pi(a ; vert ; h_t)) is the probability of taking action a given the history (h_t).
For the Bernoulli bandit, it is natural to assume that (Q(a)) follows a Beta distribution, as (Q(a)) is essentially the success probability θ in Bernoulli distribution. The value of (text{Beta}(alpha, beta)) is within the interval [0, 1]; α and β correspond to the counts when we succeeded or failed to get a reward respectively.
First, let us initialize the Beta parameters α and β based on some prior knowledge or belief for every action. For example,
- α = 1 and β = 1; we expect the reward probability to be 50% but we are not very confident.
- α = 1000 and β = 9000; we strongly believe that the reward probability is 10%.
At each time t, we sample an expected reward, (tilde{Q}(a)), from the prior distribution (text{Beta}(alpha_i, beta_i)) for every action. The best action is selected among samples: (a^{TS}_t = argmax_{a in mathcal{A}} tilde{Q}(a)). After the true reward is observed, we can update the Beta distribution accordingly, which is essentially doing Bayesian inference to compute the posterior with the known prior and the likelihood of getting the sampled data.
[begin{aligned}alpha_i & leftarrow alpha_i + r_t mathbb{1}[a^{TS}_t = a_i] beta_i & leftarrow beta_i + (1-r_t) mathbb{1}[a^{TS}_t = a_i]end{aligned}]Thompson sampling implements the idea of probability matching. Because its reward estimations (tilde{Q}) are sampled from posterior distributions, each of these probabilities is equivalent to the probability that the corresponding action is optimal, conditioned on observed history.
However, for many practical and complex problems, it can be computationally intractable to estimate the posterior distributions with observed true rewards using Bayesian inference. Thompson sampling still can work out if we are able to approximate the posterior distributions using methods like Gibbs sampling, Laplace approximate, and the bootstraps. This tutorial presents a comprehensive review; strongly recommend it if you want to learn more about Thompson sampling.
Case Study
The Slot Bandits
I implemented the above algorithms in lilianweng/multi-armed-bandit. A BernoulliBandit object can be constructed with a list of random or predefined reward probabilities. The bandit algorithms are implemented as subclasses of Solver, taking a Bandit object as the target problem. The cumulative regrets are tracked in time.
Fig. 4. The result of a small experiment on solving a Bernoulli bandit with K = 10 slot machines with reward probabilities, {0.0, 0.1, 0.2, …, 0.9}. Each solver runs 10000 steps. (Left) The plot of time step vs the cumulative regrets. (Middle) The plot of true reward probability vs estimated probability. (Right) The fraction of each action is picked during the 10000-step run.
Summary
We need exploration because information is valuable. In terms of the exploration strategies, we can do no exploration at all, focusing on the short-term returns. Or we occasionally explore at random. Or even further, we explore and we are picky about which options to explore — actions with higher uncertainty are favored because they can provide higher information gain.
Cited as:
[1] CS229 Supplemental Lecture notes: Hoeffding’s inequality.
[2] RL Course by David Silver - Lecture 9: Exploration and Exploitation
[3] Olivier Chapelle and Lihong Li. “An empirical evaluation of thompson sampling.” NIPS. 2011.
[4] Russo, Daniel, et al. “A Tutorial on Thompson Sampling.” arXiv:1707.02038 (2017).