Problem 1. A former student in this class, one Peter Merrill, had a betting problem. No, not that kind of betting problem. A real-world betting problem. Like millions of guacamole-addicted Americans, Pete's family and friends pick winners of every week's NFL games. The person with the most correct picks throughout the season wins. However, because the Bowl is THE SUPER BOWL, it is worth 5 victories to pick the winner.
Going into the big game, Pete's mom was in the lead and the next closest competitor was 4 points behind. In the old days, the folks in the pool employed a rule that the person in the lead had to reveal his or her pick before THE SUPER BOWL, but Pete's mom wisely decided that a rule change was in order and so this rule had been overturned.
Your assignment is to help Pete's mom. She's not looking for advice as to which is the better team (and the game is already over anyway). She's looking for advice as to how to bet.
The game is Denver versus Seattle and we'll assume that Denver is the favored team. Denver's chance of winning is p > 50%.
Thus under the old rules where mom had to go first, she would pick Denver and the wholly-undeserving number 2 (Wun2) in second place would pick Seattle. His chance of winning would be just 1 - p. So if p=60%, mom would win 60% of the time. Wun2 understands that his only chance of winning is if he picks a different team than mom, as a tie does him no good. And he only wins if his team also wins.
Fortunately for the Merrill family fortune, mom has changed the rules. Wun2 has to make his pick without knowing what mom has done. If Wun2 thinks mom will still pick Denver, then he will pick Seattle and mom should do an end run and pick Seattle. But if Wun2 is one step ahead of mom, he would then pick Denver and get the better team in the process.
Questions.
1. What is mixed-strategy equilibrium of this game?
2. Even if we don't know p exactly, we can be pretty confident that p is in the range of 50% to 70%. Thus what is the range of mom's chance of winning?
3. Write down a simple formula for how much better Pete's mom is as a result of the rule change. When p = 70%, how much was this change worth to her?
4. Say this mixed strategy thing is too complicated for mom and she does the following: mom flips a coin and covers the coin without seeing the result. If the coin is heads, she will bet on Denver and if tails then Seattle. What is her chance of winning? When p = 70%, how much worse is this compared to her equilibrium strategy?
5. Say that in a misguided effort to be fair, Pete explains this whole mixed-strategy business to Wun2. How much is his mom worse off if Wun2 correctly anticipates the probability that mom will be on Denver?
Problem 2.
The following summary is taken from Wikipedia
Catherine Susan "Kitty" Genovese was a New York City woman who was stabbed to death near her home in Queens in New York City, on March 13, 1964 by Winston Moseley.The circumstances of her murder and the lack of reaction of numerous neighbors were reported by a newspaper article published two weeks later;the common portrayal of neighbors being fully aware but completely unresponsive has since been criticized as inaccurate. Nonetheless, it prompted investigation into the phenomenon that has become known as the bystander effect or "Genovese syndrome" and especially diffusion of responsibility.
Genovese's killer, Winston Moseley was found guilty and sentenced to death on June 15, 1964. That sentence was later reduced to lifetime imprisonment on the grounds that he had not been allowed to argue during the trial that he was "medically insane". Moseley committed another series of crimes when he escaped from custody on March 18, 1968 and then fled to a nearby vacant home, where he held the owners hostage. On March 22, he broke into another home and took a woman and her daughter hostage before surrendering to police. Moseley, who was denied parole for a sixteenth time in November 2013, remains in prison.
The events of Genovese's death are subject to dispute. Some accounts suggest that her cries for help were heard and ignored by numerous residents in the apartment building. Other accounts, as detailed below, suggest that residents did not hear her pleas or did provide assistance or both. The exact details are unknown.
Genovese had driven home from her job working as a bar manager early in the morning of March 13, 1964. Arriving home at about 3:15 a.m. she parked in the Long Island Railroad parking lot about 100 feetfrom her apartment's door, located in an alley way at the rear of the building. As she walked toward the building she was approached by Winston Moseley. Frightened, Genovese began to run across the parking lot and toward the front of her building located on Austin Street trying to make it up to the corner toward the major thoroughfare of Lefferts Boulevard. Moseley ran after her, quickly overtook her and stabbed her twice in the back. Genovese screamed, "Oh my God, he stabbed me! Help me!" Her cry was heard by several neighbors but, on a cold night with the windows closed, only a few of them recognized the sound as a cry for help. When Robert Mozer, one of the neighbors, shouted at the attacker, "Let that girl alone!"Moseley ran away and Genovese slowly made her way toward the rear entrance of her apartment building. She was seriously injured, but now out of view of any witnesses.
Records of the earliest calls to police are unclear and were not given a high priority by the police. One witness said his father called police after the initial attack and reported that a woman was "beat up, but got up and was staggering around."
Other witnesses observed Moseley enter his car and drive away, only to return ten minutes later. In his car, he changed to a wide-brimmed hat to shadow his face. He systematically searched the parking lot, train station, and an apartment complex. Eventually, he found Genovese who was lying, barely conscious, in a hallway at the back of the building where a locked doorway had prevented her from entering the building. Out of view of the street and of those who may have heard or seen any sign of the original attack, Moseley proceeded to further attack her, stabbing her several more times. Knife wounds in her hands suggested that she attempted to defend herself from him. While Genovese lay dying, Moseley raped her. He stole about $49 from her and left her in the hallway.The attacks spanned approximately half an hour.
A few minutes after the final attack a witness, Karl Ross, called the police. Police arrived within minutes of Ross' call. Genovese was taken away by ambulance at 4:15 a.m. and died en route to the hospital. Later investigation by police and prosecutors revealed that approximately a dozen (but almost certainly not the 38 cited in the Times article) individuals nearby had heard or observed portions of the attack, though none saw or were aware of the entire incident. Only one witness, Joseph Fink, was aware she was stabbed in the first attack, and only Karl Ross was aware of it in the second attack. Many were entirely unaware that an assault or homicide was in progress; some thought that what they saw or heard was a lovers' quarrel or a drunken brawl or a group of friends leaving the bar when Moseley first approached Genovese.
To help better understand the bystander problem we are going to model the situation as a game. The process will greatly and unfairly simplify the tragic circumstances of Kitty Genovese's murder. I have shared the story because it asks the question of how crowds will act when the numbers get larger.
Assume that there are n people who witness a crime. Each person has two strategies. He or she can call for help (C) or he of she can do nothing (N). Calling for help has some cost c, which we will assume is the same for all bystanders. If one or more people call for help, the police will come. In that case, those who called get a payoff of 1 - c, while those who didn't call get a payoff of 1 (as the police have come). The point is that calling for help has a cost of getting involved, giving a statement, and the like, but it guarantees that the police will come. The cost of calling is not so large that if you knew that you were the only one who had seen the crime you would make the call: in other words, 1 - c > 0. If you don't call, there is a chance that no one else will and the police will not come in which case your payoff is 0.
Based on the wiki article, this is far from a perfect description of the Kitty Genovese story, but it is the model we are going to use for this problem.
(i) What are the pure strategy Nash equilibria to this game when n = 2 and when n = 10.
(ii) What are the mixed-strategy equilibria when n = 2 and 10? Note that you can restrict attention to symmetric mixed-strategy, those solutions in which all players employ the same strategy. For simplicity, you should assume each person calls the police with probability p.
(iii) What is the aggregate chance that the police are called? How does this chance vary with n?
(iv) How do the bystanders' expected payoffs change with the group size?
(v) What are some of the biggest weaknesses of this model?