Playing Codenames: When Frequentist Statistics Becomes Optimal

The Game

Codenames is a popular card game designed by Vlaada Chvátil published in 2015. To play this game, each turn a clue giver in each of the two competing teams announces a one-word clue and a number indicating how many words on the table are related to this clue and the guesser(s) have to guess as many words that belong to their team as possible. About 8 of the 25 words on the table belong to each of the teams. The rest of the words are neutral but guessing the black assassin loses the game. Only the clue givers know which words belong to which team. The team who has all their words guessed wins the game.

The following pictures show a game in progress where all the guessed words are covered by the team’s color (beige is neutral). The color key at the bottom is only shown to the clue givers.


The Bayes formula

The guessers guess the words in sequence, up to one more than the number specified by the clue giver. For each guess, the guessers need to find the word that has the highest probability of belonging to their team. For each candidate word, this probability is calculated as follows, using the Bayes formula.


Since each word is randomly assigned to each team, the quantity P(Guessed \, word \in Own \, team's \, words) is the same for every word. Additionally, the denominator P(Clue) is fixed for any given clue. Therefore we can simplify the equation as:


That is, given a clue, the probability that a word belongs to the guessers’ team is proportional to the probability that this clue is used given this word needs to be guessed. The left side is what we need, but is extremely hard to calculate given the strategies of the game. The right side is more straightforward. Often, the guessers have to consider the few target words for this turn as a whole set and plug into the formula.

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How AKB48 Can Revive by Balancing Exploitation and Exploration

AKB48 Group is the world’s largest girls idol group based in Tokyo, Japan. It contains six smaller constituent groups called sister groups with over 300 members in total. Naturally the popularity of each member is different which should follow a relatively symmetric unimodal distribution. Since there is no way every member can get equal and adequate opportunities to perform, a dozen or so prominent members are selected to receive half of the attention.  As the histogram below shows, however, the popularity distribution (measured by number of followers on Showroom) is very right skewed due to at least two effects: (1) the distribution of media exposure is extremely right skewed and (2) the number of years since joining the group is right skewed. We know that media exposure can shape member popularity by contrasting the popularity distribution to that of a more egalitarian group named Keyakizaka46, which has very symmetric and concentrated popularity distribution and media exposure distribution. Keyakizaka46 is able to achieve relative equality of media exposure due to its small size of 21 members (not counting under group members).

The distribution of the number of followers on Showroom is very right skewed. Source

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When Should We Replace $5 Bills With $3 Bills?

Almost all currencies in the world have denominations that start with 1, 5, or 0 and the main difference among them is whether $2, $20, $200, etc. bills are used. But prevalence does not prove efficiency. In this post, I will compare the efficiency of different bill denomination systems.

An efficient currency minimizes the cost of transaction. To simplify, we assume that the cost of transaction is proportional to the time spent on it. The cost of transaction can be dichotomized into time spent by the customer (while the cashier waits) and time spent by the cashier (while the customer waits). Each component further divides into the following categories denoted by a, b, and c:

  • Fixed transaction cost (a): time spent on taking out the wallet, opening the cash register, thinking about which bills to use (while doing nothing else), and handing and receiving the money. Avoid double counting if these jobs overlap in time or if the other party is not waiting but doing some other necessary work, such as printing the receipt.
  • Fixed cost for using bills in each denomination (b): time spent on moving the hand to reach a specific slot in the cash register; time spent on finding the place for a specific denomination in the wallet and putting all bills of this denomination on the other hand or the counter before moving on to work on the next denomination.
  • Cost for counting a bill (c).

We acknowledge that cost a can be substantially lower if no change is required from the cashier, which saves the time of passing the changes back to the customer. However, most transactions involve taxes and multiple items that lead to the usage of coins (in the US). Most of the time the customer does not have the exact amount of coins or does not want to pay any coins. Secondly, for the same reason, charges are usually not psychologically convenient numbers (e.g. $5, $20). Nice numbers can make buying decisions easier but they don’t show up in the payment. Therefore we can assume that there are always changes involved and a is constant and can be entirely dropped from this analysis.

Continue reading “When Should We Replace $5 Bills With $3 Bills?”

How to Recover Deleted Coursera and Berkeley Courses from

In June 2016, deleted 472 open online courses as it migrated from an old system to a new system; from March to August 2017, UC Berkeley made all 350+ courses on Webcast.berkeley private. Fortunately, most of these courses have been archived by the Archive Team. This post provides the instructions for downloading and opening the archived courses.

Recovering UC Berkeley courses

Recovering Berkeley courses is easy. Go to this link:

and find the course you want to watch and click on the link to in the right column. You can watch the videos online or download them. You can find archived course descriptions in this page or search in Berkeley’s academic guide.

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The Evolution of Risk Preferences

In this post, I explore the evolutionarily adaptive risk preferences under various conditions.

Fair wager model

To understand the long term effects of risk aversion vs risk neutrality, we consider an evolutionary player with 10 apples initially who repeatedly makes a series of wagers that gets one apple half of the time and pays one apple half of the time. The wager is considered “fair” since the expected payoff is 0. In a population, every player is presented with the same wagers but the results of the wagers are independent or idiosyncratic. The player stops playing when he loses all the apples and dies. The wager repeats 10000 times in a short time and we record the number of apples owned at the end, which is proportional to the population size or fitness. Three thousand simulations are run and the distribution of the number of apples is plotted in figure 1.1.

Figure 1.1

As plotted in figure 1.1, in the long run the player almost surely dies. The distribution is severely skewed right. The single period expected payoff and long run expected payoff are both 0 apples because starvation does not change expected future payoff since whether wagering or dead, the expected payoff remains 0. Can we say evolution disfavors this kind of wagers since the extinction rate is high? Without additional assumptions, the answer is no. If individuals in a population either always accept this wager or decline this wager, then after many generations more than 99% of the lineages will be from the risk averse ones who decline the wager, which makes risk aversion seem like the dominant strategy. However, if counting the number of surviving individuals, the two risk preferences are equally successful by having equal total population sizes since the expected payoff of both risk preference equals 0. We will use this method to define fitness in the rest of the article and therefore we can assume what evolution maximizes is the long run expected number of offsprings produced. Assuming every offspring is the same, an organism selected by maximizing the long run fitness should maximize the number of his children (and kins) in his lifetime.  Continue reading “The Evolution of Risk Preferences”

The Five Goals of Lawmaking and Three Approaches of Combining Them

Making and executing ideal laws with the right penalty is an optimization process with debatable objectives. But at least we can list the five components of the objectives and three components of the costs in an attempt to model lawmaking. Later we can look for the prevailing approaches to combine these goals in ideal lawmaking and determine the origins of these approaches.

Goals of lawmaking

1. Change future behavior of criminals

This is often thought of as the goal of imprisonment. We would like criminals to have limited opportunities to commit another crime while in prison and less likely to commit crimes after they are released, which we attempt to achieve through enrichment programs in prison.

2. Compensate the victims

Victims can be financially compensated through fine or psychologically compensated through apology or by knowing that the criminals will be punished.

3. Deter people from committing crimes

When the punishment is severe enough, it is no longer worth it to commit a crime even when the probability of getting caught is small.

4. Improve the perceived fairness of society

Most people would like to live in a just society where people who harm others without permission are punished.

5. Make the criminals better people

Criminals are people too and many believe that the society should be responsible to help them by pulling them out of the wrong path and teaching them what’s right.

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The Deterioration of Coursera

Coursera is the biggest massive open online course (MOOC) platform founded in 2012 with over 1900 courses currently being offered. The website today is nothing like what it used to be in 2015, especially after moving to a new system in mid-2016. It’s not surprising that Coursera remains the alpha in the MOOC industry since it’s an oligopolistic market with universities mostly sticking to the same platform over time. But it’s mind-blowing to know that highly competent people can create a great platform and then destroy it a few years later.

So what happened? For each good feature they added, they removed five other great features. After the 2016 update, course providers are no longer allowed to place a promotional video in the course description and old videos are removed. Users could no longer click on professor names to see their profiles, which took Coursera a year to fix this problem. Professors should deeply appreciate the reappearance of their profiles, because not only are links to university profiles gone, the catalog of all university providers has also been deleted recently.

Coursera has redefined MOOC to be Monthly Open Online Course, as most of the courses now are running a new session every month, and even allowing students to switch sessions if they miss any deadline, helping procrastinating students strengthen their habit of ignoring deadlines. With so many ongoing sessions at once, course sizes have decreased dramatically and the discussion forums are now for roam, as students stare at the blank screen and wonder why nobody complies when the professor asks them to write down their thoughts in the forums. Forum posts are never kept from session to session, strangling the only way left to revive the forums. The collaboration with Coursera is gone too, and the Coursera experience is now predominantly solitary. Continue reading “The Deterioration of Coursera”