2007-12-22

The classical prisoner's dilemma

The Prisoner's Dilemma was originally framed by Merrill Flood and Melvin Dresher working at RAND in 1950. Albert W. Tucker formalized the game with prison sentence payoffs and gave it the "Prisoner's Dilemma" name (Poundstone, 1992).

The classical prisoner's dilemma (PD) is as follows:

Two suspects, A and B, are arrested by the police. The police have insufficient evidence for a conviction, and, having separated both prisoners, visit each of them to offer the same deal: if one testifies for the prosecution against the other and the other remains silent, the betrayer goes free and the silent accomplice receives the full 10-year sentence. If both remain silent, both prisoners are sentenced to only six months in jail for a minor charge. If each betrays the other, each receives a five-year sentence. Each prisoner must make the choice of whether to betray the other or to remain silent. However, neither prisoner knows for sure what choice the other prisoner will make. So this dilemma poses the question: How should the prisoners act?

The dilemma can be summarized thus:


Prisoner B Stays Silent Prisoner B Betrays
Prisoner A Stays Silent Each serves six months Prisoner A serves ten years
Prisoner B goes free
Prisoner A Betrays Prisoner A goes free
Prisoner B serves ten years
Each serves five years

The dilemma arises when one assumes that both prisoners only care about minimizing their own jail terms. Each prisoner has two and only two options: either to co-operate with his accomplice and stay quiet, or to defect from their implied pact and betray his accomplice in return for a lighter sentence. The outcome of each choice depends on the choice of the accomplice, but each prisoner must choose without knowing what his accomplice has chosen.

In deciding what to do in strategic situations, it is normally important to predict what others will do. This is not the case here. If you knew the other prisoner would stay silent, your best move is to betray as you then walk free instead of receiving the minor sentence. If you knew the other prisoner would betray, your best move is still to betray, as you receive a lesser sentence than by silence. Betraying is a dominant strategy. The other prisoner reasons similarly, and therefore also chooses to betray. Yet by both defecting they get a lower payoff than they would get by staying silent. So rational, self-interested play results in each prisoner being worse off than if they had stayed silent. In more technical language, this demonstrates very elegantly that in a non-zero sum game a Nash Equilibrium need not be a Pareto optimum.

Note that the paradox of the situation lies in that the prisoners are not defecting in hope that the other will not. Even when they both know the other to be rational and selfish, they will both play defect. Defect is what they will play no matter what, even though they know fully well that the other player is playing defect as well and that they will both be better off with a different result.

The "Stay Silent" and "Betray" strategies are also known as "don't confess" and "confess", or the more standard "cooperate" and "defect."

One experiment based on the simple dilemma found that approximately 40% of participants cooperated (i.e., stayed silent).[1]

2007-12-21

Prisoner's Dilemma 囚徒困境

A real experience about the Game Theory of Prisoner’s Dilemma.

The story happened on last Friday. I traveled to a Vendor somewhere in China from Hong Kong. I have been visiting this vendor many times in the past month. The traveling time from custom building to the vendor is about 30 minutes by taxi. But the return trip will be a bit difficult because there is no direct taxi back to the custom building. At the area where the vendor located, there is a different kind of taxi (green in color). People have to take the green taxi first to another district and change another kind of taxi (red in color) to the custom building. The taxi operation is somewhat like that in Hong Kong. Taxi in green is only able to travel within the New Territories while Taxi in red is allowed to travel to everywhere in Hong Kong.


It is almost impossible to get a Taxi in red near the vendor’s location. So normally, visitors will call up a private car with driver to drive back to the custom building after finishing my work there This is so call "white label cab" and same as in Hong Kong it is against the local traffic law. However, the White Label Cab is more convenient and requires less time to go to the custom building. If I take the hard way, riding on green taxi and then change to red taxi, the trip may take up to 2 hours in order to arrive the custom building.


Last Friday, I called up a White Label Cab driver to pick me up at 18:00 near the vendor’s place as usual. After getting on the cab, I slept during the journey until the cab nearly arrived the custom building. I was waken up by a guy pounding on the car window. And he said he was a police officer of Traffic Control Unit. He asked me to leave the car and to present my Identity Card to him. Then, he took my Identity Card with him. He asked me how much I paid for the ride. Immediately, I was totally woke up and sensed that the Police Officer wanted to make a case of illegal ride of White Label Cab. So I told the office I did not need to pay for anything for the ride. This was just a friendly drive of my friend to the custom building. At that time, I still thought that this should be a small traffic case and would not cause too much trouble. In Hong Kong, this should be just a fix penalty to the White Label Cab driver and no penalty on passenger.


But unfortunately, this happened in China. The police officer took me back to a police car and directed the driver to follow him to a Police Station nearby. On the way to the Police Station, I kept on complaining that I was in a hurry back and they got no reason to hold me up. One of the Police Officers told me that once I revealed how much I paid for the ride, I would be released right away. At that time, I was just thinking about this "deal" and assessing how trustworthy the Police Officers could be. When we arrived the Police Station, the car driver and I were held in custody in different rooms inside the police station. This was the first time I was "arrested" in China and had to stay in an examination room alone to face cross-examination.


Then, the Game Theory about Prisoner’s Dilemma applied. A new group of 3 Police Officers with a video recorder and a booklet they called it a confession note. The chief officer was fierce and made very clear he would not believe one word I told them. He asked me how much I paid the driver again and again. I said it was only a friendly ride. He said it was not true and asked how I met the driver, and asked me to write down the name and where was the driver’s hometown. In addition, if I confessed that I hired a White Label Cab and signed my name on the confession note, I would be freed right away. If not, I had to prepare to stay in the examination room for the whole night. Certainly, I only knew the name of the driver and did not know anything about his life history. At that time, I was struggling in my mind. If I told the truth that I hired the White Label Cab, I may be able to get out of it but the driver would be penalized. The other possibility would be, if I admitted the car hiring, the driver and I would be penalized together. Then I recalled the Game Theory, especially the Prisoner’s Dilemma. The best solution of that kind of separate cross-examination should be that: both of the parties do not admit anything no matter what the alligator said.


After I wrote down the name of the driver, phone number and his hometown that I really didn’t know (I wrote down mine as his instead ^____^), the officer took my note to match with what the driver wrote down. Then, the officer became even more fierce and said there was nothing match on the note with the driver’s and accusing me that I was telling lies to them. Calmly, I told the officer, that was what I knew from the driver and I had no responsibility to check it was right or not. I kept on my statement that I needed to go as soon as possible. The officer seemed very dissatisfied and left the room with my Identity Card. He threw his last word to me that I had to stay there for the night as I did not tell the truth. I raised my voice and said I already told them what I knew and I re-iterated I needed to go with the intension to put pressure on them.


After the Police Officers left the room, I evaluated my situation in my mind and tried to figure out if my strategy was correct. After a deep thought, I decided to continue my denial approach as this should be the best way out according to the theory of Prisoner’s Dilemma. And I anticipated that the police officer would make me stay there for a long while in order to force me to admit. I was planning to phone to my Lawyer for help if I had to wait too long.


To my surprise, after the first group of officer left, in a minute or two, there came another Police Officer in uniform. He was much tender (playing the "good crop" role) and told me another story. He said the police was against those White Label Cab simply because of concerning the safety of the passengers. He told me that there was a guy from Hong Kong who was murdered few days ago because he got on a White Label Cab and not knowing where the driver took him. Then, he asked me again how much I paid for the driver. I insisted my story and re-iterated that I was in a hurry to go, in order to give him pressure. And I told this officer that, if the traffic and taxi arrangement was better in the area I came from, I did not have the need to ask the help from my friends to give me a ride. With this chance, I complained to the officer the traffic there should be improved and not to accuse people of riding White Label Cabs. He seemed a bit stuck with what I said and eventually asked me if I was willing to take any legal responsibility for what I told them. I avoided the word "any", but just said to him I told the truth and not sure the legal responsibility of the truth meant. Finally he walked away with my Identity Card.


Again I was not sure how long would it take. Another surprise to me, after a minute or two, another police officer appeared in the room and gave me back my Identity Card and told me to leave without signing anything. I picked up my Card and rushed to the gate. On the way, I met the driver and he shouted at me that we should have dinner after that. At first, I planned to leave the police station and headed to the custom building by taking another taxi outside. But as the driver made a dinner appointment to me, I would like to check with him what was going on his side. So I waited outside the Police Station because I also still wanted to pay him for the ride.


After a short while, the driver was also released from the Police Station and picked me up on the roadside. I said no need to buy me dinner and asked him to find a safe place near the custom building that I could pay him. At time, he was terrified and he showed tremendous gratefulness on me. After we sat down in a restaurant, he told me what would be the result if I admitted the case. He would be penalized for at least RMB30,000, his car would be took away and his driving license would be banned for 2 years. He said if not with my help, he would be dead this time and he insisted not to receive any payment from me. However, I said he drove me to the destination although bad things happened, yet still he should receive the money for his service. He also told me that the traffic police officers were recently very active in catching White Label Cabs simply because there were some sharing of interest between the Police and Taxi companies. As White Label Cabs took some business from the taxis, the taxi companies were putting pressure on the police to get rid of other drivers who ate up some of their profits.


Finally, the driver received his payment and I headed back to the custom building. The whole happenings just caused 20 to 25 minutes of delay. By the theory of Prisoner’s Dilemma, eventually I got through the difficulties.

2007-12-12

Serenity Prayer

祈求上蒼賜我
寧靜的心接受無法改變的現實
勇氣去改變可以改變的事情
且有智慧明白兩者的分別

The Serenity Prayer is a prayer written by Reinhold Niebuhr on July 1, 1943. Early in World War II, the prayer was printed on cards and distributed to the troops by the U.S.O. However, the prayer is reported to have been used before that date in Twelve-step programs for those recovering from hurts, habits, addictions and hang-ups.



The Serenity Prayer

God grant me the serenity
to accept the things I cannot change;
courage to change the things I can;
and wisdom to know the difference.

Living one day at a time;
Enjoying one moment at a time;
Accepting hardships as the pathway to peace;
Taking, as He did, this sinful world
as it is, not as I would have it;

Trusting that He will make all things right
if I surrender to His Will;
That I may be reasonably happy in this life
and supremely happy with Him
Forever in the next.

Amen.

From Reinhold Niebuhr