Game Theory on kenji.blog

Why did building a new road somehow make traffic worse?: Braess's Paradox

Thu, 10 Sep 2026 21:00:00 +0900

Morning rush hour. You are frustrated by the roads that are jammed every day, but good news has arrived. “To eliminate traffic congestion, the city planning department has built the latest shortcut road!” Everyone must have hoped that they could sleep a little longer starting tomorrow.

However, the next day, when the new road opened, instead of making things better, it caused a much worse traffic jam than before, and everyone’s commute time became longer.

This is not an urban legend or an administrative failure. It is a famous phenomenon in network theory called “Braess’s Paradox”, which was mathematically proven by the German mathematician Dietrich Braess in 1968.

The Paradox Model: 4,000 Commuters

Let’s check with a simple mathematical model why the phenomenon of “everyone slowing down even though the number of roads has increased” occurs.

There are 4,000 drivers heading from the starting point (residential area) to the goal point (business district). Initially, there were only two routes (upper route and lower route) as follows.

Upper Route: Go through a narrow road $A$, and then through a wide highway $B$.
Lower Route: Go through a wide highway $C$, and then through a narrow road $D$.

The “narrow road” gets congested as the number of cars increases, so the travel time takes “the number of running cars $\div 100$” minutes. The “wide highway” never gets congested no matter how many cars come, and always takes “45 minutes”.

Travel Time [Before Road Construction]

The drivers are smart, so they try to choose a faster route even slightly. As a result, the 4,000 people are evenly divided into the upper route (2,000 people) and the lower route (2,000 people).

Upper Route Travel Time: $\frac{2000}{100}$ minutes (narrow road) + $45$ minutes (highway) = $65$ minutes
Lower Route Travel Time: $45$ minutes (highway) + $\frac{2000}{100}$ minutes (narrow road) = $65$ minutes

No matter which route is chosen, the travel time stabilizes at “65 minutes” for everyone.

The Trap of the Shortcut Road

Now, suppose the mayor has built a “dream ultra-high-speed bypass that allows you to travel from Waypoint 1 to Waypoint 2 in 0 minutes (instantly)”.

The drivers have gained a new route option. A driver standing at the starting point thinks like this. “It’s better to use the narrow road A than to use the highway C (45 minutes). Because even at worst, if all 4,000 people chose A, it would only take 40 minutes (4000/100).”

Therefore, all 4,000 people head to “Narrow Road A”. When they arrive at Waypoint 1, they think again. “It’s better to go through the new bypass (0 minutes) and use the narrow road D than to use the highway B (45 minutes). Because even if everyone goes through D, it’s 40 minutes at worst.”

Therefore, all 4,000 people head to “Narrow Road D” through the “New Bypass”.

Travel Time [After Road Construction]

As a result of everyone making “the fastest (rational) choice for themselves”, everyone ends up taking the same route (A → New Bypass → D).

Let’s calculate the travel time.

Narrow Road $A$: $\frac{4000}{100} = 40$ minutes
New Bypass: $0$ minutes
Narrow Road $D$: $\frac{4000}{100} = 40$ minutes
Total: $80$ minutes

Surprisingly, despite the creation of a convenient new shortcut, everyone’s commute time worsened from “65 minutes” to “80 minutes”.

You might think, “Why doesn’t at least one person use the back road (the old route)?”, but if one person chooses the old highway route (45 minutes + 40 minutes = 85 minutes), it will be even slower than the current 80 minutes, so no one tries to change their route. In game theory, this is said to have reached a “Nash Equilibrium”. As a result of everyone taking the optimal action for themselves, it has fallen into the worst outcome as a whole.

Real-world Examples

Braess’s Paradox is not just an armchair theory; it has been observed multiple times in real-world urban traffic and network systems.

1969 Stuttgart, Germany: A new road was built to relieve traffic congestion, but the congestion worsened. Eventually, when the new road was closed off, traffic flow improved.
1990 New York: When “42nd Street”, a mecca of traffic congestion, was completely closed off for an Earth Day event, contrary to the expectations of traffic experts, the overall congestion in Manhattan was dramatically relieved.
Communication Networks: The same phenomenon can occur in internet routing and power grids. The moment a new cable or line is added, data packets can concentrate on the “perceived optimal shortest path”, sometimes causing the entire network to go down.

Braess’s Paradox beautifully illustrates the dilemma of complex societies: “a collection of rational individual choices (egoism)” does not necessarily lead to “an optimal outcome for the whole”. Sometimes, “taking away choices (freedom)” can be to the benefit of everyone.

Newcomb's Paradox: Can You Beat a Superbeing Who Foresees the Future?

Thu, 10 Sep 2026 08:00:00 +0900

1. The Ultimate Choice Game

An alien with superintelligence who calls himself “Omega” appears before you. Omega is a master of analyzing human behavior and possesses the terrifying ability to “predict what choice a subject will make next with almost 100% accuracy.” In past experiments, Omega’s predictions have never been wrong.

Omega places two boxes in front of you.

Box A: A transparent box. It contains exactly 100,000 yen.
Box B: An opaque box. It contains either 100 million yen or is empty (0 yen).

Omega tells you to choose one of the following two actions.

Choice 1: “Take both boxes”: You get both the 100,000 yen in Box A and the contents of Box B.
Choice 2: “Take only Box B”: You get only the contents of Box B. You must give up the 100,000 yen in Box A.

Hearing just this, anyone would decide to “Take both boxes”. However, Omega adds a terrifying “rule”.

[Omega’s Rule]

“Yesterday, I already predicted ‘which choice you will make’ today and set the contents of Box B. If I predicted you would greedily choose to ‘Take both boxes’, I left Box B empty. If I predicted you would not be greedy and choose to ‘Take only Box B’, I put 100 million yen in Box B.”

Now, you must make a choice. Should you “Take both boxes”? Or should you “Take only Box B”?

graph TD Omega["Omega's Prediction
(already completed yesterday)"] Omega -->|Predicts 'Take both'| BoxB_Empty["Box B is empty (0 yen)"] Omega -->|Predicts 'Take only Box B'| BoxB_100M["Puts 100 million yen in Box B"] You["Your Choice
(today)"] You -->|Choice 1: Take both| Result1["Box A (100k) + Box B contents"] You -->|Choice 2: Take only Box B| Result2["Box A (0 yen) + Box B contents"] BoxB_Empty -.-> Result1 BoxB_100M -.-> Result2

2. Two Clashing “Perfect Logics”

This problem was devised by physicist William Newcomb in 1969 and published by philosopher Robert Nozick. As soon as it was published, the opinions of brilliant mathematicians and philosophers around the world were split right down the middle, causing a massive controversy.

This is because there is an “absolutely irrefutable, perfect logic” for either choice.

Logic 1: The argument of the “Take only Box B” camp (Expected Value Maximization)

“Omega’s prediction accuracy is almost 100%, right? Then we should trust Omega based on past data. If I choose to ‘Take both’, Omega has foreseen it, and the result is just 100,000 yen. If I choose to ‘Take only Box B’, Omega has foreseen it, and the result is 100 million yen. Even a fool knows whether they want 100,000 yen or 100 million yen. Therefore, I should absolutely ‘Take only Box B’!”

This thinking is based on “Expected Utility Theory,” which straightforwardly believes in past statistical data and expected values.

Logic 2: The argument of the “Take both boxes” camp (Dominant Strategy)

“Wait a minute. Omega predicted and put the contents in Box B ‘yesterday’, right? That means, at this moment, the contents of Box B are already determined to be either ‘contains 100 million yen’ or ’empty’, and it will never change.

Pattern 1: If Box B already contains 100 million yen, choosing to ‘Take both’ gives me 100.1 million yen, and choosing ‘Only B’ gives me 100 million yen. Pattern 2: If Box B is already empty, choosing to ‘Take both’ gives me 100,000 yen, and choosing ‘Only B’ gives me 0 yen.

In either pattern, choosing to ‘Take both’ absolutely gets me 100,000 yen more! Whatever I choose now, Omega’s actions yesterday cannot be rewritten by a time machine. Therefore, I should absolutely ‘Take both boxes’!”

This thinking is based on the “Dominant Strategy” in game theory, which says to “choose the option that is advantageous to you regardless of what action the opponent takes.”

3. Do you believe in “Free Will”?

“The camp that takes only Box B” and “The camp that takes both”. After hearing both arguments, which did you think was right?

Actually, to this day, there is no “mathematically perfect single correct answer” to this paradox. Because at the root of this problem lies humanity’s greatest philosophical question: “Determinism vs. Free Will”.

Those who answered “Take only Box B” (Determinists)

People who make this choice subconsciously accept “Determinism (everything in the future of this world is determined from the beginning)”. The fact that Omega can predict the future with 100% accuracy means that your current decision was not chosen by “your free will,” but rather “you were already destined to choose that way since yesterday by the physical laws of the universe and the movement of neurons in your brain.” Since the future cannot be changed, this view holds that it is most rational to follow Omega’s prediction and ride the “destiny of taking only Box B.”

Those who answered “Take both boxes” (Free Will advocates)

People who make this choice subconsciously believe in “Free Will (you can carve out the future through your own choices)”. Because they believe that “regardless of Omega’s prediction yesterday, I can change my choice with my own will right now,” they take the action to “add 100,000 yen at this very moment, regardless of the already determined contents of the box.” Even if the result is that Omega predicted it and the box is empty, they possess the logic to accept that “it cannot be helped because it is the result of taking logically correct action.”

4. Time Travel and the Collapse of Causality

What makes Newcomb’s paradox even more complicated is the reversal of “Causality (cause and effect)”.

In the common-sense world we live in, “My choice today (cause)” creates “Tomorrow’s result”.

However, in Omega’s game, It seems that “My choice today (cause)” determines “Yesterday’s Omega’s action (result)”. A “backward causality” occurs, where future actions determine past facts.

If a “perfect predictor” like Omega exists in the universe, even our common sense that “time flows from the past to the future” collapses.

5. Conclusion: A Thought Experiment that Uncovers Human “Rationality”

Which box will you open?

More than half a century has passed since this paradox was presented, but in surveys of philosophy and economics, opinions are beautifully split half-and-half between the “Take both” camp and the “Take only B” camp. And interestingly, both camps genuinely believe that “the opponent’s logic is completely bankrupt and foolish”.

“What is a rational judgment?” No matter how much economics and mathematics develop, in the end, it comes down to the philosophy of “how humans perceive this world.” Newcomb’s paradox is a magnificently mean and beautiful thought experiment that confronts us with the limits of logic.

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.