Another very short movie: 7 minutes.
http://www.youtube.com/watch?v=RpjHSiQLPmA
In this movie, we see that the Jew and Arab reaction depend on the others reaction. For example, the Jew reaction depends on the Arab reaction and on the Nazis reaction. Since Game Theory is when one's decision depends on other's decision, and you have different outcomes, we can say that this movie has Game Theory.
By the way, excellent movie! If I can give my personal opinion.
Thursday, December 11, 2008
Wednesday, December 10, 2008
More for The Dark Knight
During a special interrogation by Batman, Joker reveals that Rachel and Harvey have been kidnapped and placed in 2 different places. He interchanges the location of the places when he tells Batman and Batman rushes off thinking he will save Rachel. When he reaches, he ends up saving Dent just as he hears Rachel die (through a phone link) at the other end.
Joker predicted how Batman would think (he would choose to save Rachel) and interchanged the places.
Joker predicted how Batman would think (he would choose to save Rachel) and interchanged the places.
The Dark Knight and Game Theory
The Dark Knight movie has a lot og game thoery. Here I chose one scene to describe:
The Story
The Joker's final act as criminal mastermind and agent of nihilism (or, seemingly, to show Gotham city that we are all Homo Economicus when the structure of the game forces us to be) involves two ferries filled with people. The first ferry is filled with normal, law abiding citizens while the second ferry is filled with the population of Gotham Prison. The Joker, doing so without prior knowledge of the passengers and city officials, wired the ships with powerful explosives such that their explosion would destroy the entire ship and everyone aboard. No single individual is allowed to escape. Each ship is given a detonator for the other ferry. The use of the detonator saves the ship while killing everyone aboard the opposing ship. Thus, if any member of Ship A pushes the detonator, then Ship B is destroyed and all of Ship A is saved. Additionally, if either ship fails to use the detonator to destroy its opponent, then both ships will be destroyed by the Joker.
Solving this game is pretty straightforward and (detonation, detonation) becomes the dominant strategy as, at best, cooperation is weakly dominated. Thus, homo economicus and politicus have a very clear strategy to, without fail, destroy their opponents, and in so doing, both will ships will be destroyed.
The game appears to be pseudo-sequential or, perhaps, a series of simultaneous game with a finite end. The Joker gives both ferries 30 minutes in which they can detonate the other side. Even with this complication, the outcome should be the same and both actors ought to choose detonation at the first node. Using backwards deduction, both players recognize that their opponent will choose to detonate in the final iteration even if there are some gains to short-term cooperation. This effect cascades backwards to the initial decision node and mutual detonation occurs to prevent receiving the sucker's payout of cooperating while the opponent defects.
Decision Rules
Beyond this, it appears that the decision making process for both ships is different. In the ship containing prisoners, the decision to detonate becomes decentralized and any one actor willing to grab the detonator could do so. While the armed guards gives the opposite impression of clear authoritarianism, Decentralization becomes apparent as the time moves on. Thus, decentralized decision making should lead to the optimal play as any single individual among the 500 or more sub-actors should have a preference for survival.
On the civilian ship, the decision mechanism becomes a simple majority vote. When the votes are aggregated, the decision to detonate the other ferry is chosen at a rate of almost 3 to 1. Yet, there is no executive to carry out the decision and the majority will does not prevail as no single sub-actor is willing to push the button. The civilians act rationally as long as they, individually, are not too involved in carrying out a potentially morally reprehensible act.
Morality?
Perhaps social norms mattered for the actors in the game? In the second game, we can assume that there are some social benefits from being a moral agent; however, being moral is not as beneficial as being alive. Since survival trumps morality, we get the second game:
As Yev Kirpichevsky notes in his comment, the pure strategy equilibria are {cooperate-detonate} and {detonate-cooperate} with a Mixed Nash Equilibrium of playing cooperation and detonation with a probability of .5 for both players with this specification (this would change depending on how we parametrize the value for being moral and the value for surviving). Perhaps, then, we saw the cooperate, cooperate cell (with probability .25) in the movie and if we watched the movie infinite more times, we would see a nice distribution where cooperate-cooperate occurs 25% of the time, both ships defect 25% of the time, and only one ship explodes 50% of the time. I plan on seeing the movie again, and I will let you know if the outcome of this scene changes.
The Story
The Joker's final act as criminal mastermind and agent of nihilism (or, seemingly, to show Gotham city that we are all Homo Economicus when the structure of the game forces us to be) involves two ferries filled with people. The first ferry is filled with normal, law abiding citizens while the second ferry is filled with the population of Gotham Prison. The Joker, doing so without prior knowledge of the passengers and city officials, wired the ships with powerful explosives such that their explosion would destroy the entire ship and everyone aboard. No single individual is allowed to escape. Each ship is given a detonator for the other ferry. The use of the detonator saves the ship while killing everyone aboard the opposing ship. Thus, if any member of Ship A pushes the detonator, then Ship B is destroyed and all of Ship A is saved. Additionally, if either ship fails to use the detonator to destroy its opponent, then both ships will be destroyed by the Joker.
Solving this game is pretty straightforward and (detonation, detonation) becomes the dominant strategy as, at best, cooperation is weakly dominated. Thus, homo economicus and politicus have a very clear strategy to, without fail, destroy their opponents, and in so doing, both will ships will be destroyed.
The game appears to be pseudo-sequential or, perhaps, a series of simultaneous game with a finite end. The Joker gives both ferries 30 minutes in which they can detonate the other side. Even with this complication, the outcome should be the same and both actors ought to choose detonation at the first node. Using backwards deduction, both players recognize that their opponent will choose to detonate in the final iteration even if there are some gains to short-term cooperation. This effect cascades backwards to the initial decision node and mutual detonation occurs to prevent receiving the sucker's payout of cooperating while the opponent defects.
Decision Rules
Beyond this, it appears that the decision making process for both ships is different. In the ship containing prisoners, the decision to detonate becomes decentralized and any one actor willing to grab the detonator could do so. While the armed guards gives the opposite impression of clear authoritarianism, Decentralization becomes apparent as the time moves on. Thus, decentralized decision making should lead to the optimal play as any single individual among the 500 or more sub-actors should have a preference for survival.
On the civilian ship, the decision mechanism becomes a simple majority vote. When the votes are aggregated, the decision to detonate the other ferry is chosen at a rate of almost 3 to 1. Yet, there is no executive to carry out the decision and the majority will does not prevail as no single sub-actor is willing to push the button. The civilians act rationally as long as they, individually, are not too involved in carrying out a potentially morally reprehensible act.
Morality?
Perhaps social norms mattered for the actors in the game? In the second game, we can assume that there are some social benefits from being a moral agent; however, being moral is not as beneficial as being alive. Since survival trumps morality, we get the second game:
As Yev Kirpichevsky notes in his comment, the pure strategy equilibria are {cooperate-detonate} and {detonate-cooperate} with a Mixed Nash Equilibrium of playing cooperation and detonation with a probability of .5 for both players with this specification (this would change depending on how we parametrize the value for being moral and the value for surviving). Perhaps, then, we saw the cooperate, cooperate cell (with probability .25) in the movie and if we watched the movie infinite more times, we would see a nice distribution where cooperate-cooperate occurs 25% of the time, both ships defect 25% of the time, and only one ship explodes 50% of the time. I plan on seeing the movie again, and I will let you know if the outcome of this scene changes.
Sunday, December 7, 2008
The Game Theory of Penalty Kicks
As a Brazilian person, I can not forget the most popular sport in Brazil: soccer. It is not that I care much for it, but at least all Brazilian male do it.
The six dimensions of Game Theory classification for Penalty
- Simultaneous Game
- One-shot
- Zero-sum
- Full information / partial
- Rules Fixed
- Cooperatives Agreements possible? No!
Depending of the championship's rules, the game can not end equal, and each team has an equal number of chances to make penalty kicks (PK) in a shootout. In 2002 World Cup, Korea advanced to the semifinals after knocking off Spain on PKs. In 1998, France could never have won the tournament had it not edged Italy in PKs in a quarterfinal match. And, of course, had Roberto Baggio not shanked his PK in 1994, the Italians could have been world champions themselves, but then Brazil won the World Cup for the fourth time. So, the ability to make penalty kicks (and stop them if you are a keeper) is tremendously important. For Ignacio Palacio-Huerta, that ability is the focus of an exhaustive study that reveals how keepers and shot-takers alike deal with penalty kicks. But Palacios-Huerta doesn't analyze player tendencies in order to help a particular team, his interest in soccer has led him to write several papers about how motivations, risk, and reward influence decision-making on the pitch.
Penalty kicks stood out to him because they are a rare real-life manifestation of two-person zero-sum games. A penalty kick is a situation in which the shot-taker either scores or doesn't score based on simultaneous actions taken by both the shot-taker and the keeper. Simplistically, both the shot-taker and keeper (the "players") must decide whether to aim right or left, without knowing the direction that the other will aim (the shot-taker aims with the ball, while the keeper aims with his body). Both players generally do better going to one of the two sides, and, logically, will choose to play their strong side more frequently than their weak side. However, neither player can always choose his strong side, because then the other player will know where to aim. Therefore, each player must decide how frequently to play each side, so as to maximize his expected payoff (for the shot-taker, the probability of scoring; for the keeper, the probability of preventing a score). Players should decide these frequencies such that their expected payoff will be the same whether they aim right or left; and each time a player aims right or left, the move should be random (unpredictable).
To see if professional soccer players were following classical game theory, Palacios-Huerta watched over one thousand penalty kicks taken in the highest professional leagues of England, Italy, and Spain. What he found was that, while different in their success levels, almost all frequent penalty-kick players were superb game theorists, choosing to aim right or left with appropriate frequencies.
Why kicking to the center could be a good idea? How he would advise teams to approach penalty kicks? The interview has been edited for clarity:
Gelf Magazine: Are these really zero-sum games? The outcome for the team could simplistically be +1 or -1 (for goal or no goal, if you don't consider the current score or the time left in the game), but these may not be the reward/punishment for the kicker and keeper specifically. They may be looking to increase their fame, pay, etc., and not specifically looking to get or prevent a goal. Also, depending on the score, the kicker could be a hero if he gets the goal and the team wins (+1), but the keeper only has a small punishment (-0.2) if he misses the small shot he had of preventing the goal.
Ignacio Palacios-Huerta: This is very interesting. I think that: For the most part they are zero-sum. The empirical evidence and all the statistical analysis are consistent with PK being zero-sum. One advantage of zero-sum games is that everything depends ONLY on the fact that the interests of the players are exactly opposed to each other. In the case of PK, one definitely wants to score and the other wants to stop (no-score). This is the case regardless of whether there are other things associated with "score" or "no-score" (fame, embarrassment, etc.), and also regardless of the different degrees of importance of a penalty kick (i.e., whether it is a terribly important penalty or whether it will not affect the final outcome (win or lose) of the match), one still wants to score and the other wants no-score.
GM: The reason more players don't shoot to the center on PKs is because it would be embarrassing to get such a kick blocked. Do you agree?
IPH: No, I do not. One can readily make exactly the opposite argument, namely that it is a great honor to score shooting to the middle, and not a big deal to have it stopped (rather than an embarrassment to have it stopped and not a big deal to score).In fact, I think that in some sense it is a great honor. The most famous penalty shot (and I think the first one) to the middle was taken by Panenka in 1976. It is so famous that it has a name: when a penalty is shot softly to middle, say, 1 meter or 1.5 meters above the ground, it is said that the penalty was shot a la Panenka. It is very risky but the fame payoffs is great.
IPH: No, I do not. One can readily make exactly the opposite argument, namely that it is a great honor to score shooting to the middle, and not a big deal to have it stopped (rather than an embarrassment to have it stopped and not a big deal to score).In fact, I think that in some sense it is a great honor. The most famous penalty shot (and I think the first one) to the middle was taken by Panenka in 1976. It is so famous that it has a name: when a penalty is shot softly to middle, say, 1 meter or 1.5 meters above the ground, it is said that the penalty was shot a la Panenka. It is very risky but the fame payoffs is great.
GM: In Slate's coverage of your study, Gianluigi Buffon and Zinedine Zidane are praised for being unpredictable and thus having a great understanding of game theory.
IPH: Actually, if you look at my paper, the vast majority are unpredictable.
GM: But as the New York Times points out, neither Zidane nor Buffon are particularly effective compared to other keepers or kickers. Do you think it's possible that their focus on being unpredictable has made them less effective? (On a related note, do you think that any players actually worry about being random?)
IPH: I have talked to many players, and my sense is that they are not really thinking of being unpredictable. At least, I do not think that Zidane or Buffon are really focusing more than others... In general, some players have some gut feelings about where to shoot or where to move, some do not really know where to shoot or move, some change their mind in the last millisecond. If they start thinking a lot about it they will probably stop being random.
IPH: I have talked to many players, and my sense is that they are not really thinking of being unpredictable. At least, I do not think that Zidane or Buffon are really focusing more than others... In general, some players have some gut feelings about where to shoot or where to move, some do not really know where to shoot or move, some change their mind in the last millisecond. If they start thinking a lot about it they will probably stop being random.
GM: You write that it would make sense for players to be unpredictable in their PK patterns.
IPH: Yes, this is part of the equilibrium strategy. In equilibrium, (1) the scoring rate should be the same across the different choices that they have (in the simplest case, across left and right); and (2) they should be unpredictable, the same way it is unpredictable which side a coin is going to land.
GM: But do you think that, practically, any teams analyze the patterns of players they may face when it comes to PKs (besides obvious giveaways like always going right)?
IPH: In my experience, they analyze very little. It is more at the level of individual players (goalkeepers and designated kickers) than at the level of teams. Some players do keep written records, but by and large many, and I think most, do not. Interestingly enough, though, what players have is a terrific PK memory. If you ask them, they remember very well what they did and what their opponents did, in many, many penalties even far back in time, sometimes going back years. Somehow, and probably unconsciously, they have those records in their brain.
GM: Often, the keeper chooses the correct side but still gets beaten. How does game theory apply in these cases?
IPH: In my experience, they analyze very little. It is more at the level of individual players (goalkeepers and designated kickers) than at the level of teams. Some players do keep written records, but by and large many, and I think most, do not. Interestingly enough, though, what players have is a terrific PK memory. If you ask them, they remember very well what they did and what their opponents did, in many, many penalties even far back in time, sometimes going back years. Somehow, and probably unconsciously, they have those records in their brain.
GM: Often, the keeper chooses the correct side but still gets beaten. How does game theory apply in these cases?
IPH: No problem at all. On the contrary, this is part of the game. As I mentioned above, the "payoffs" are the probabilities that a goal will be scored or not for each combination of strategies. These probabilities are simply constructed using the observed frequencies (e.g., of all the Left-Left [keeper and player both play to the left] 59 percent were goals and 41 weere no-goal).
GM: But are there cases in which a player is so skilled to one side that he should not try to be random?
IPH: I have never seen any one like this, and I do not think any like this has ever existed. Perhaps the closest was Alan Shearer, who shot a lot to the right and little to the left (but, as predicted, he had—statistically speaking—the same success rate on either side). I think that no player like this ever existed because when one gets too close to ALWAYS using one strategy all the time, the goalkeeper will tend to ALWAYS go that way, in which case the kicker will always have the other side open and should then begin shooting to the other side ... I think something like this may have happened to Alan Shearer in his career...
Even if I am not a big fan of soccer, who doesn't remember this moment in the 1994 World Cup, when Baggio lost his penalty, and gave Brazil the World Cup leadership?
Source and full article:
Youtube
GBS Honor Code - a fair distribution of information
The Honor Code of the Goizueta Business School establishes a level playing field where no one student can take unfair advantage over others. The Code is supported by a set of procedures and guidelines, known as “The Honor System”, which is designed to promote a productive and open academic environment. As such, it is essential that each student understands his or her role in the Honor System.But then we have to clarify what would be a fair information advantage and an unfair information advantage. Because if all students had the same exactly information in the moment they take a test, all of them would receive the same grade, and it is not what happen. It does not happen like this because students can have fair advantage over others, according to the honor code.
An information fair advantage is, for example:
- a student who studies more than another student previous the test
- a student who has work experience in the subject, for example, accounting
- a student who looks for the professor previous to the test to better understand some points
An information unfair advantage is, for example:
- a student who shares information about the test content with another student during the test
- a student who looks in the material class if the instructions says that the student can not do that
- a student who looks for the professor in the moment of the test to better understand some points
Sources:
GBS website
Saturday, December 6, 2008
The Wave - psicological analysis to be part of a group
The Wave is one of the best movies I have ever seen. It is a 45 minutes movie, and if you have no idea what I am talking about, I strongly recommend you to watch it. But you don't need to watch, I wrote about it.
If the Part 1 below does not work, use the "full video" option, below:
Part 1:
Part 1:
If the Part 1 above does not work, use the video below. If the image to watch the video also does not work, go to the link:
The full video:
The Wave is about a real teaching experiment by Ron Jones that took place in a high school history class in Palo Alto, California, in 1967.
The plot revolves around history teacher, Mr. Ben Ross, who cannot answer the question of why the Germans allowed Adolf Hitler and the genocidal Nazi Party to rise to power, acting in a manner inconsistent with their own pre-existing moral values. The only way he can see to answer the question is to start an experiment that shows the students what it may have been like in living in Nazi Germany.
Ben starts by having his history class sit up straight and obey his commands by, at first, standing at attention beside their desks and having to say "Mr. Ross..." before asking questions or answering questions he asked them. After seeing the students' reactions toward the experiment, he decides to continue it the next day by creating a salute, a symbol and addressing three mottoes he made up: "Strength through discipline, Strength through community, Strength through action." He calls this movement "The Wave". At first, students are skeptical about The Wave, but after seeing how everyone becomes equal, and that the stress of making choices are lifted, the class falls into The Wave, and begins to recruit others into it. Robert, the class reject, seems to have changed the most due to The Wave - his physical appearance becomes neater and the students grow to accept him more.
Laurie, a student in Mr. Ross' class, starts to think that The Wave is having too much of an impact. A huge majority of the school is in The Wave, and its members attack students who refuse to join. Using her influence as the School Newspaper Editor, Laurie releases an entire issue of The Grapevine dedicated to showing the dangers of The Wave. While some thank her, especially teachers and parents, others do not. Laurie's boyfriend David, who has been in The Wave since the beginning, tries to get her to stop bad-mouthing The Wave. He eventually shoves her to the floor and then realizes what harm The Wave has done.
After talking with Laurie and David, as well as his wife, Christy, who is also a teacher at the school, Ross realizes that The Wave has taken a turn for the worse, and is determined to stop it. However, he is determined to do so in a way that communicates the lesson he intended for The Wave to teach in the first place. He calls a Wave meeting in the auditorium and requests that only Wave members be present. They gather in a similar fashion to the Nazi rallies, even equipped with banners and armbands emblazoned with the Wave logo.
Ben tells The Wave members that they are about to see the leader of the whole organization and that he is going to speak to all of them on television to create an international Wave Party for Youths. Everyone is shocked when Mr. Ross reveals that there is no leader, and that there is no international Wave Party. However, Mr. Ross tells the audience that if there were a leader, it would be the man on the projection screen - Adolf Hitler. He explains how their obedience led them to act like Nazis.
The shocked students drop all their Wave-branded trinkets and items, and slowly leave the gym. As Ben turns to leave, the one person who really flourished in the Wave, Robert Billings, is standing alone, upset that The Wave ended. During The Wave, he was finally accepted as an equal, no one picked on him, he had friends, but his new found social status is now worthless without The Wave. Mr. Ross tries to cheer him up by commenting on his tie and suit, and they walk out together.
Sources:
Wikipedia
Youtube
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