Psychology on kenji.blog

The Prisoner's Dilemma: Why Do We Make Choices Where "Everyone Loses"?

Thu, 10 Sep 2026 03:00:00 +0900

1. The Ultimate Choice: Remain Silent or Betray?

You and your accomplice friend have been caught by the police on suspicion of a certain crime. The two of you are placed in separate interrogation rooms and cannot communicate with each other at all.

Because the police have not fully solidified the evidence, the prosecutor offers each of you and your friend the following “plea bargain”.

If both “remain silent (cooperate)”: Due to insufficient evidence, both of you will only get 1 year in prison.
If you “confess (betray)” and your friend “remains silent”: You, who cooperated with the investigation, will be found not guilty (immediate release), but your friend will take all the blame and get 10 years in prison. (And vice versa)
If both “confess (betray)”: Since both admitted to the crime, the sentence is slightly reduced and both get 5 years in prison.

Now, what would you do? Will you “remain silent (cooperate)”? Or will you “confess (betray)”?

2. Analysis Using a Payoff Matrix

Let’s organize this situation into a “payoff matrix” used in game theory. The numbers in the squares represent (your prison years, your friend’s prison years). A minus indicates a loss (prison years).

You \ Friend	Remain Silent (Cooperate)	Confess (Betray)
Remain Silent (Cooperate)	(-1, -1)	(-10, 0)
Confess (Betray)	(0, -10)	(-5, -5)

Viewed objectively, the optimal action the two should take is clear. If both “remain silent”, the total sentence is only 2 years (-1 and -1). This is the “Pareto optimal” state that maximizes the overall benefit.

However, if you are a “rational human being trying to maximize only your own benefit”, a completely different conclusion is drawn.

3. Why Is “Betrayal” a Rational Choice?

Let’s follow the thought process of deciding your own action by predicting the action of your “friend” in the other room.

Case 1: If you predict your friend will “remain silent”

If you also “remain silent”, 1 year in prison.
If you “confess”, not guilty (immediate release). $\rightarrow$ Being not guilty is better, so “confess (betray)” is optimal.

Case 2: If you predict your friend will “confess”

If you also “remain silent”, 10 years in prison.
If you “confess”, 5 years in prison. $\rightarrow$ 5 years in prison is better, so again, “confess (betray)” is optimal.

Have you noticed? No matter what action the other person takes, “confessing (betraying)” is always more advantageous for you. In game theory, this is called a “dominant strategy”.

Your friend is placed in exactly the same situation and thinks rationally in exactly the same way, so “confessing” also becomes the dominant strategy for your friend.

As a result, the two rational thinkers will always both choose to “confess (betray)”. The resulting outcome is 5 years in prison for both (-5, -5), which is nearly the worst outcome overall. Even though cooperating (remaining silent) would have resulted in only 1 year in prison, pursuing individual rationality leads to mutual loss.

graph TD Start["Start of Choice"] --> Logic_You["Your Rational Thinking"] Start --> Logic_Friend["Friend's Rational Thinking"] Logic_You -->|If the other is silent, confessing is better
If the other confesses, confessing is better| Betray_You["You choose to confess (betray)"] Logic_Friend -->|If the other is silent, confessing is better
If the other confesses, confessing is better| Betray_Friend["Friend chooses to confess (betray)"] Betray_You --> Result["Result: Both confess (-5, -5)"] Betray_Friend --> Result Ideal["Ideal: Both remain silent (-1, -1)"] -.->|Individual rationality blocks
reaching this| Result style Result fill:#ff9999,stroke:#333,stroke-width:2px style Ideal fill:#99ff99,stroke:#333,stroke-width:2px

This state, “where neither has an incentive to change strategy as a result of predicting the other’s action (nothing more can be done)”, is called a “Nash Equilibrium”, named after the master of game theory, John Nash.

The most terrifying point of the Prisoner’s Dilemma lies in the fact that “Pareto optimal (the best result for the whole)” and “Nash equilibrium (the end point of individual rationality)” do not coincide.

4. The “Prisoner’s Dilemma” Hidden in Everyday Society

The Prisoner’s Dilemma is not just a quiz. Many problems occurring in our society can be explained by this mathematical model.

1. Price Competition (Price War)

Two rival companies are selling a similar product for 1000 yen. If both companies keep the 1000 yen price (cooperate), both can gain high profits. However, succumbing to the temptation to “make it slightly cheaper than the competitor (betray) and monopolize customers”, both companies start a price war. As a result, the product becomes 500 yen, and both companies suffer without making a profit (mutual betrayal).

2. Environmental Issues and Greenhouse Gases

Countries around the world promise to “reduce CO2 emissions (cooperate)”. This is the optimal solution for the entire Earth. However, if only one’s own country “ignores emission limits and operates factories (betrayal)”, only its own economy can grow rapidly. Conversely, if other countries betray but one’s own country strictly follows the rules, only one’s own country will suffer a huge economic loss. As a result, every country fears being outsmarted and chooses to betray, and the global environment is destroyed.

3. Doping Problems in Sports

Ideally, all athletes should refrain from doping (cooperate). However, due to paranoia that “the opponent might be doping” or the temptation that “I can win if I am the only one doping”, they choose doping (betrayal). As a result, they fall into the worst situation where everyone competes full of drugs while ruining their health.

5. Is There a Solution? The “Tit for Tat” Strategy

In a single transaction, “betrayal” always becomes the rational choice. However, when this becomes a “game repeated over and over with the same opponent (Iterated Prisoner’s Dilemma)”, the situation changes dramatically.

In the 1980s, political scientist Robert Axelrod held a tournament matching up computers programmed with various strategies. Among complex strategies gathered from scholars around the world, such as “Always Betray”, “Betray Randomly”, and “Forgive the Opponent”, the one that won with an overwhelming score was the simplest “Tit for Tat” strategy.

The rules of the Tit for Tat strategy are just these:

Always “cooperate” on the first move.
From the next move onwards, just copy “the action the opponent took” in the previous move.
- If the opponent cooperated last time, cooperate this time.
- If the opponent betrayed last time, retaliate by betraying this time.

This strategy is strong because it has four characteristics: “Never betray first (Nice)”, “Immediately punish when betrayed (Retaliatory)”, “Immediately forgive if the opponent changes their attitude (Forgiving)”, and “Simple structure that is easy for the opponent to understand (Clear)”.

graph LR Start["1st Time: Cooperate unconditionally"] --> Round2 Round2["Observe opponent's action"] -->|Opponent cooperated| Act_Coop["Cooperate as well"] Round2 -->|Opponent betrayed| Act_Betray["Betray as well (Retaliate)"] Act_Coop --> Round2 Act_Betray -->|If opponent repents
and returns to cooperation| Act_Coop

In human relationships and international society as well, if a long-term relationship is assumed, by sharing a rule like the “Tit for Tat” strategy—“basically cooperate, but penalize betrayal”—we can overcome the prisoner’s dilemma and build cooperative relationships.

6. Conclusion: The Value of “Trust” Taught by Mathematics

The Prisoner’s Dilemma mathematically proved that “human selfish rationality” can sometimes plunge the entire society into the depths of misery. The individual rationality of “wanting to be the only one who profits” or “not wanting to be outsmarted” ultimately invites a result (Nash equilibrium) that strangles one’s own neck.

At the same time, however, game theory also teaches us that as long as the condition of “the relationship continuing long-term” is met, “trusting and cooperating with each other” is the most rational strategy that ultimately maximizes one’s own profit as well.

The next time you wonder, “Should I cheat just a little bit for myself?”, try to remember this payoff matrix of the Prisoner’s Dilemma. Pursuing immediate profit through “rational betrayal” might be the most irrational choice in the long run.

The Missing Dollar Riddle: Learning Logical Thinking and Accounting Basics from an Intuition-Deceiving Math Paradox

Thu, 10 Sep 2026 00:00:00 +0900

1. Introduction: Why Are We Deceived by Simple Addition?

In the world, there exist strange problems that can completely bug out the human brain using only lower elementary school level “addition” and “subtraction”, without relying on advanced calculus or complex topology. Among these, the most famous worldwide and the one that has troubled many people is “The Missing Dollar Riddle”.

At first glance, it seems like an ordinary everyday scene, a story about a bill issue at a restaurant (or hotel). However, just by following the calculations a little, “$1” suddenly vanishes from the world.

In this article, we will take up this famous math paradox (more accurately, a paradox-style trick question) and thoroughly dissect why our intuition is deceived and where the logical pitfalls lie, from three perspectives: mathematics, cognitive psychology, and double-entry bookkeeping (accounting).

2. Posing the Problem: The Missing Dollar Riddle

First, please read the following story. And if you have a pen and paper at hand, please try following the calculations together.

[!QUESTION] The Missing Dollar Riddle (Story) One day, three travelers came to a small hotel. The front desk clerk at the reception said, “A room for three is $30 total for one night." > The three travelers each took out $10 from their wallets, paid a total of $30 to the clerk, and headed to their room.

After a while, the hotel manager came by and told the clerk: “Today is a campaign day, so that room is only $25. Go return $5 immediately.”

The clerk headed to the guest room holding $5 in bills. However, he thought to himself on the way: > "It's hard to split $5 equally among 3 people. If I secretly take $2 and return the remaining $3, it fits perfectly at $1 per person.”

So, the clerk hid $2 in his pocket, lied to the travelers saying "You get a $3 refund from the campaign," and returned $1 to each person.

Now, here is the problem.

The travelers initially paid $10 each and later got back $1 each, so the amount they actually paid is $10 - $1 = $9.

The total amount paid by the 3 travelers is $9 × 3 people = $27.

Meanwhile, the clerk has the secretly pocketed $2 in his pocket.

If you add the $27** paid by the travelers and the **$2 held by the clerk, it becomes 27 + 2 = $29.

Initially, the travelers definitely paid “$30". > However, according to the current calculation, there is only "$29”.

Where in the world did the remaining $1 disappear to?

How about it? The more you read, the more your brain might get confused thinking, “There is definitely $1 missing!". The calculation formulas themselves are as simple as `9 × 3 = 27` and `27 + 2 = 29`, which even elementary schoolers can understand. And yet, for some reason, it doesn't add up to the initial $30.

From the next chapter onward, let’s unravel the trick behind this strange phenomenon.

3. The Gap Between Intuition and the Right Answer: Why Does the Brain Bug Out?

The thought process that many people fall into when hearing this problem is as follows:

graph TD A["Initial State: Guests pay $30"] --> B["Refund Process: Manager returns $5"] B --> C["Fraud: Clerk steals $2"] C --> D["Guests' Final Cost: $9 × 3 people = $27"] D --> E["Mysterious Calculation: Guests' cost $27 + Clerk's $2 = $29"] E --> F["Question: Doesn't match initial $30! $1 vanished!"] style E fill:#ff9999,stroke:#333,stroke-width:2px style F fill:#ff4444,color:#fff,stroke:#333,stroke-width:4px

The true identity of this paradox lies in a clever word trick (Framing Effect) of “adding things that shouldn’t be added”.

The Core of the Fallacy: The Meaningless Calculation of “27 + 2”

Look closely again at the following part at the end of the problem text:

If you add the $27** paid by the travelers and the **$2 held by the clerk, it becomes 27 + 2 = $29.

Actually, this calculation of “27 + 2” itself is a logically completely meaningless calculation. This is because the “final amount paid by the travelers ($27)" already includes the "amount secretly pocketed by the clerk ($2)”.

The breakdown of the $27 paid by the travelers is as follows:

Amount in the hotel cash register: $25
Amount pocketed by the clerk: $2
Total: $27

In other words, adding the clerk’s $2 on top of the $27 means you are “Double Counting” the clerk’s $2.

If you want to correctly match it up with the initial “$30”, you need to add together the “amount paid by the guests” and the “amount returned to the guests”.

Final amount paid by the guests: $27 (Register $25 + Clerk $2)
Amount returned to the guests: $3
Total: 27 + 3 = $30

Calculated this way, it becomes obvious that not a single dollar has disappeared.

4. Mathematical Clarification: Strict Proof by Equations

For those who are not satisfied with just a verbal explanation, let’s prove the cash flow using strict mathematical formulas.

Let’s define the overall movement of money with variables.

$ P_{initial} $ : Total amount initially paid by guests (30)
$ C_{hotel} $ : Final amount received by the hotel (manager) (25)
$ R_{total} $ : Refund amount handed to the clerk by the manager (5)
$ R_{guest} $ : Final refund amount received by the guests (3)
$ S_{waiter} $ : Amount pocketed by the clerk (waiter) (2)

From the initial cash flow, the following equation holds:

$$ P_{initial} = C_{hotel} + R_{total} \quad \cdots (1) $$

($30 = $25 + $5)

The $5 returned by the manager is divided between the guests and the clerk's pocket. $$ R_{total} = R_{guest} + S_{waiter} \quad \cdots (2) $$ ($5 = $3 + $2)

Substitute equation (2) into equation (1).

$$ P_{initial} = C_{hotel} + (R_{guest} + S_{waiter}) \quad \cdots (3) $$

($30 = $25 + $3 + $2)

Here, we define the “final amount paid by the guests” mentioned in the problem as $ P_{final} $. This is the initial amount paid minus the amount returned to the guests.

$$ P_{final} = P_{initial} - R_{guest} \quad \cdots (4) $$

($27 = $30 - $3)

Let’s transpose $ R_{guest} $ to the left side from equation (3).

$$ P_{initial} - R_{guest} = C_{hotel} + S_{waiter} \quad \cdots (5) $$

From equation (4) and equation (5), the following truth is derived:

$$ P_{final} = C_{hotel} + S_{waiter} \quad \cdots (6) $$

(Guests’ final payment $27 = Hotel sales $25 + Clerk’s theft $2)

The trick of the problem lies in the fact that it tries to add $ S_{waiter} $ ($2), which is supposed to be included in the right side, once again to $ P_{final} $ ($27) on the left side. In other words, the calculation formula induced by the problem text is as follows:

$$ P_{final} + S_{waiter} = (C_{hotel} + S_{waiter}) + S_{waiter} $$ $$ 27 + 2 = (25 + 2) + 2 = 29 $$

This number “29” is a fictional value with absolutely no physical or economic meaning, simply being “hotel sales + clerk’s theft × 2”. This is the mathematical true identity of the illusion that makes it seem like “$1 disappeared”.

5. The Accounting Perspective: Shattering the Paradox with Double-Entry Bookkeeping

For those who are still not convinced (or feel intuitively foggy) even with mathematical equations, you can perfectly visualize this mystery using the concept of “Double-Entry Bookkeeping”, which has been used in the business world for over 500 years.

The basic principle of double-entry bookkeeping is that “Debit” and “Credit” always match. Let’s use this to make a journal entry of the movement of money.

Transaction 1: Guests pay $30

This is the initial state from the hotel’s perspective.

Debit (Increase in Assets)	Credit (Increase in Liabilities/Equity)
Cash: $30	Deposits (or Sales): $30

Transaction 2: Manager hands $5 to clerk, records $25 as sales

Since the room charge was changed to $25, $5 is given to the clerk for “refund”.

Debit	Credit
Deposits: $30	Sales: $25 Clerk (Cash): $5

Transaction 3: Clerk’s action ($3 refund and $2 embezzlement)

This is the most important part. We record the destination of the $5 cash held by the clerk.

Debit	Credit
Refund to Guests: $3 Embezzlement Loss: $2	Clerk (Cash): $5

Final Integrated State of Balance Sheet (B/S) and Profit & Loss (P/L)

As a result of the whole process, we summarize where the cash is and under what name.

pie title Final location of the initial $30 (Assets side) "Hotel Register (Sales $25)" : 25 "Guests' Wallets (Refund $3)" : 3 "Waiter's Pocket (Embezzlement $2)" : 2

[Confirmation of Final State]

Source of funds (Guests’ expense): $30
Location of funds (Result):
- $25 in the hotel register
- $2 in the waiter’s pocket
- $3 with the guests
- Total = 25 + 2 + 3 = $30

Looking at it through the accounting “principle of matching debits and credits (T-accounts)”, the “amount paid by the guests $27 (expense)" is a "decrease on the asset side", and the act of adding the "waiter's stolen $2 (movement on the asset side)” to it is nothing but an impossible mistake of “mixing and adding debits and credits” according to accounting standards. In the business world, if an accountant reported the calculation “27 + 2 = 29” to management, it is a logical failure on the level that they would immediately be fired or suspected of accounting fraud.

6. The Cognitive Psychology Perspective: Why Do We Accept “27+2=29”?

Why do many people unconsciously accept a calculation formula that is mathematically and accountingly wrong, thinking “Hmm, I see”? That involves powerful cognitive biases built into the human brain.

1. The Bug in Mental Accounting

Behavioral economist Richard Thaler (Nobel Prize winner in Economics) proposed that humans unconsciously categorize money in their heads (“Mental Accounting”). At the end of the problem text, the “guests’ expense ($27)" and the "waiter's obtained money ($2)” are presented under the same category of “money”. The brain just extracts the “amount numbers (27 and 2)” and easily performs addition, ignoring the direction of the vectors—whether it is “money paid (negative)” or “money held (positive)”.

2. Framing Effect (Information Framework)

This is the effect where people’s decision-making and judgment change depending on how information is presented. The clever part of the problem text is **“setting the initial number of $30 as the goal"**. After being shown the calculation "27 + 2 = $29”, the brain unconsciously tries to forcefully link it to the goal (anchoring) of “it should become the original $30”. It is designed to cause intense cognitive dissonance (discomfort and confusion) by making you compare numbers that shouldn’t originally be compared, and an error of “1” arises there.

3. The Magic of Storytelling

Humans are better at understanding “stories” than mathematical formulas. While simulating the movements of the characters (guests, manager, waiter) in your head, your working memory fills up, and the cognitive resources to verify the logical validity of the final equation are depleted. The exact same technique as “misdirection”, where a magician guides the audience’s gaze to succeed in a trick, is used in this word problem.

7. History and Similar Problems of “The Missing Dollar Riddle”

This kind of paradox has existed since ancient times and has been passed down in various variations across eras and borders.

Origin of the Paradox

The exact origin of this problem is unknown, but it became widely known in America in the 1930s. At the time, it was called the “Bellboy paradox”, and the amount settings varied. It is said to reflect the mass psychology during the Great Depression in America, where the whereabouts of a mere “$1” was a major concern.

An Even More Advanced Derivative: The Missing Square Puzzle

Applying this “verbal deception” to “shapes (geometry)” is the “Missing square puzzle”, which is introduced in another article on this blog. It is an intuition bug where, when rearranging shape parts that should have the same area, a hole of 1 square (area) somehow disappears. This also uses the cognitive limit that “the human eye cannot detect slight distortions (differences in slope) of straight lines”.

8. Lessons for the Real World: What Should We Learn from the Paradox?

“The Missing Dollar Riddle” has deep lessons that are a pity to end as just a drinking party joke or a child’s quiz.

The Ability to Doubt the “Given Framework (Premise)” When we make decisions in everyday business or investments, aren’t we swallowing whole the presentation materials or sales pitches presented by someone saying “Adding this number and this number results in this”? Even if the calculation result is correct (27 + 2 is definitely 29), critical thinking asking “Does setting up that equation itself even make logical sense in the first place?” is indispensable.
The Absoluteness of Cash Flow Accounting fraud in corporate accounting and loss concealment in complex financial derivatives can be said to be highly sophisticated “Missing Dollar Riddles”. Even if one makes it look like there is profit by adding and subtracting numbers without substance, tracing the “movement of cash (cash flow)” from the root will inevitably expose the contradiction. Exactly when things feel complex, it is necessary to return to the basics of “Where did the money come from, and where did it go?”.

9. Conclusion: The $1 Was Never Missing from the Start

Finally, I would like to conclude by presenting the most concise and powerful answer to this paradox.

“The guests paid a total of $27; $25 went into the hotel register, and $2 went into the waiter's pocket. The calculation is perfectly correct. The calculation formula trying to forcefully return to the initial $30 is the very source of all the confusion.”

No matter how much our brains evolve, they are very easily deceived by the combination of a “plausible story” and “simple addition”. However, by using the powerful tools of mathematics and logic (equations and double-entry bookkeeping), we can sever that illusion and see through to the truth.

Next time, if a friend poses “The Missing Dollar Riddle” with a smug face, definitely try to reply coolly with this deep knowledge in the background, “The vectors of the numbers to add and the numbers to subtract are wrong!”