Slot Machine Probability Problems

Slot Machine Probability Problems – Reinforcement learning provides a great boost to many applications, especially in e-commerce for discovering and anticipating customer behavior, including where I work as a data scientist, Wayfair. A popular way to model problems for RL algorithms is as a “multi-bar thief”, but I’ve always thought the term was unnecessarily difficult, considering it should be a helpful description. First of all “one-armed bandit” is 100-year-old slang, and second, the image of a slot machine with multiple arms to draw is a weird one.

Modern slot machines can have different buttons to press, that at least pretend to give different odds, but the best example would be several plans in the casino, some are “loose” and some are “too. ” When I walked into the Celadon City Game Corner, in the 2004 Gameboy Advance game Pokémon FireRed, and saw rows of slots all with different odds, I knew I had found the ideal “real-world” version of this example – and just useful. application of reinforcement learning.

Slot Machine Probability Problems

Celadon Game Corner: A pit of vice, corruption, and lost souls. (Screenshot by Author that is fair use on the basis of education, scholarship, and research)

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And I mean practical! How can I win 4000 coins to buy Ice Beam or Flame Thrower abilities, which I will need to fight the Elite Four??

I trained a maintenance training agent, using the Thompson model, to tell me which device to check next, and, finally, what to get the hell out of. I call it MACHAMP: Multi-Armed Coin Holdings Amplifier Made for Pokemon.

Given a set of possible actions (the “arms” of a multi-role group – in this case different devices to try), Thompson’s model has an imperfect trade-off in exploration vs exploitation to find the best action, by trying actions that are often promised, etc. getting a more detailed estimate of their profit possibilities. At the same time, he is also randomly suggesting others from time to time, if one of them is the best after all. At each step, knowledge of the system, in the form of posterior probability distributions, is updated using Bayesian logic. The simplest version of the one-armed bandit problem involves Bernoulli trials, where there are only two possible outcomes, reward or no reward, and we are trying to determine which action has the highest probability of reward. .

As a demonstration of how Thompson sampling works, imagine that we have 4 slot machines, with a 20%, 30%, 50% and 45% chance of payout. Then we can compare how the gambler finds that Slot 3 is the best one. Here and in the rest of the notebook, I start from the code that Lilian Weng wrote for the best tutorial (everything in

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At the beginning, we do not know anything about the probabilities of the devices, and assume that all the values ​​for their real game probability are possible, from 0% to 100% (depending on the problem, the choice of the Bayesian prior can be .a poor assumption, as I discuss below).

One step of the solver involves randomly sampling from the posterior probability distributions of each device, and trying the best one (this is Thompson’s sampling algorithm), then adjusting these distributions based on whether there is a game.

We can see from the graph of the estimated probabilities that a win for machine 4 has made us more optimistic about that machine – we think that the higher guesses for the winning probability are more.

After running for 100 simulated draws of four machines, we can see that it is honed on the best estimates of probabilities.

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And after 10000 trials we are still more confident that 3 has a high probability of winning, because we sampled 3 much more than the others. We also check 4 very much to be sure, but 1 and 2 we learn very quickly very bad and therefore we check more often – we have more accurate and less confident estimates of their game probabilities, but we he doesn’t care.

19 possible slot machines in the Celadon game corner, which pay out in coins that can be used to buy TM (Pokémon abilities) and Pokémon that are not available anywhere else. Three wheels rotate, and you press a button to stop them at once, with the goal to line three of the same picture, or at least a combination that starts with a cherry.

This gives 6 coins, or “just enough to keep these holders drunk” (Screen by Author that is fair use based on education, scholarship, and research)

The best jackpot is triple 7s, for 300 coins. How do I know that devices have different odds? Because a game writer told me so.

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Before going to something as ridiculously complicated as the Thompson sampling MAB solver, I looked online for another tip for beating the casino. Maybe because it’s a pretty old game (I get to them when I get to them) the information is sparse and sometimes contradictory:

So I decided that I would play by mashing the “stop” button as quickly as possible without any attention to the visuals, record only whether it was a win (of any size) or a loss, and let the Thompson sampling, by MACHAMP, do guide my choice of device to try next.

To start the predictor, though, I decided to try every device four times, and use the results to generate posterior probabilities. With only four pulls it is difficult to draw any conclusions about which are good or bad machines, because the probability distributions are very extreme – when they are not identical.

Since the overlaps make it very difficult to read the statistics for individual devices, I instead consider confidence intervals for each device: the range of possible values ​​within a certain probability, in this case 80%. It is easy to choose which machines are possible in either 4/4 or 0/4 games, and how unlikely that a 0/4 machine will turn out to be better than a 4/4 machine. However, there is a large amount of uncertainty, and it is not clear that it is not close to picking the best machine.

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Then I start using the vendor to recommend which device will run next. It was very interesting to feel the balance of discovery and exploitation as the algorithm sent me from one device to another, with games or no games. After each trial I update MACHAMP with the reward I received (0 or 1), and then ask for a recommendation for which to try next.

The trail I made around the casino would have looked pretty wacky, like Billy wandering around in the old Family Circus comic. Of course I’m fighting my efforts to stay with the machine that seems to win to the exclusion of all others, and instead randomly leave the “hot” ones to try the seemingly unpromising ones that I didn’t think of by age. People don’t think in Thompson-sampling the best ways!

I stopped after 1000 pulls of the slot machine levers, and look what I learned. First of all, there is a bias in which machines I have sampled, and it comes down to the most promising machines based on achievements.

The extent of sampling is expressed in the final confidence intervals, which are generally available for devices that appear to be worse.

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), but I can see the ones that are among the best, and how they differ from the ones that are probably the worst, e.g. machine 5, which returns exactly 0 profit in 8 draw.

If all I cared about was getting an accurate count on each machine, I could just spread all 1000 draws correctly across 19 machines, 52 draws each, but this would have led to a lot of lost coins as I continued. machines that are profitable. clearly a loser, what is called regret. Although to save time I don’t track my winnings, or even count the jackpots, after 1000 draws, MACHAMP has increased my bankroll from 120 coins to 3977.

. It has one of the best estimated game probabilities (42.1%), but, also important, in a narrow confidence interval, thanks to all the times I tried (119): I can be confident that it is specific among the best.

I did another 1000 pulls only on machine 9, both to test these calculations in practice, and to make that money. (Also, it’s election day, and it’s better to hit repost on the news…) Out of all 1119 pulls I got 37.7% of the time, which is noticeably lower than the MACHAMP estimate – although it’s just within the 80% range trust term. I think the algorithm is biased against the 50% guess, as a result of the previous uniform (starting with the guess that all values ​​between 0 and 100% are correct). Knowing what I know now, that these machines may not pay more than 40%, I could have started with a different precedent that would allow me to get more accurate estimates with the same number of tests.

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For this period of exploiting device 9, I have started to keep track of my holdings over time, and

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