ASSIGNMENT 4
Deadline:15:00PM/Thursday/NOV.6th
1.The Battle of the Bismarck Sea This game is set in the South Pacific in 1943. Admiral Imamura must transport Japanese troops from the port of Rabaul in New Britain, across the Bismarck Sea to New Guinea. The Japanese fleet could either travel north of New Britain, where it is likely to be foggy, or south of New Britain, where the weather is likely to be clear. U.S. Admiral Kenney hopes to bomb the troop ships. Kenney has to choose whether to concentrate his reconnaissance aircraft on the Northern or the Southern route. Once he finds the convoy, he can bomb it until its arrival in New Guinea. Kenney’s staff has estimated the number of days of bombing time for each of the outcomes. The payoffs to Kenney and Imamura from each outcome are shown in the box below. The game is modeled as a“zero-sum game.” For each outcome,
a.Is there any dominant strategy for Kenney?
b.Is there any dominant strategy for Imamura?
c. Is there any pure strategy Nash equilibrium? Explain your result.
d.Is there any mixed strategy Nash equilibrium?
2. The Hawk-Dove Game One fascinating and unexpected application of game theory occurs in biology. This problem is based on an example developed by the biologist John Maynard Smith to illustrate the uses of game theory in the theory of evolution. Males of a certain species frequently come into conflict with other males over the opportunity to mate with females. If a male runs into a situation of conflict, he has two alternative “strategies.” A male can play “Hawk” in which case he will fight the other male until he either wins or is badly hurt. Or he can play “Dove,” in which case he makes a display of bravery but retreats if his opponent starts to fight. If an animal plays Hawk and meets another male who is playing Hawk, they both are seriously injured in battle. If he is playing Hawk and meets an animal who is playing Dove, the Hawk gets to mate with the female and the Dove is defeated and disappears from the scene. If an animal playing Dove meets another Dove, they both stay around for a while. Eventually the female either chooses one of them or gets bored and wanders off. The expected payoffs to each of two males in a single encounter depend on which strategy each adopts. These payoffs are depicted in the box below.
Neither type of behavior, it turns out, is ideal for survival: a species containing only hawks would have a high casualty rate; a species containing only doves would be vulnerable to an invasion by hawks or a mutation that produces hawks, because the population growth rate of the competitive hawks would be much higher initially than that of the doves.
Now answer the following questions by filling in the blanks:
[1] Is there any dominant strategy for the Hawk? For the Dove? Any dominated strategy for the Hawk? For the Dove?
[2] Any Nash equilibrium of pure strategy?
[3] While wandering through the forest, a male will encounter many conflict situations with another male. Suppose all of the other males in the forest act like Doves. Any male that acted like a Hawk would find that his rival always retreated and would therefore enjoy a payoff of _ on every encounter.
[4] If a male acted like a Dove when all other males acted like Doves, he would receive a payoff of __.
[5] If you know that you are meeting a Dove for sure, it pays to be a ______.
[6] If all the other males acted like Hawks, then a male who adopted the Hawk strategy would be sure to encounter another Hawk and would get a payoff of _____.
[7] If instead, this male adopted the Dove strategy, he would again be sure to encounter a Hawk, but his payoff would be __.
[8] If everyone plays Hawk, it would be profitable to play ______.
[9] In general, not every male is a Hawk. Suppose the fraction of a large male population that chooses the Hawk strategy is p. Then if one acts like a Hawk, the fraction of one’s encounters in which he meets another Hawk is about p and the fraction of one’s encounters in which he meets a Dove is about 1?p. Therefore the average payoff to be ing a Hawk when the fraction of Hawks in the population is p, must be _______________
[10] Similarly, if one acts like a Dove, the probability of meeting a Hawk is about p and the probability of meeting another Dove is about (1 ? p). Therefore the average payoff to being a Dove when the proportion of Hawks in the population is p will be ----------------------
[11] Write an equation for p such that in such population the payoff to Hawks is the same as the payoffs to Doves. -----------------------------
[12] Solve this equation for the value of p such that at this value Hawks do exactly as well as Doves. This requires that -------------------.
[13] Use a graph to explain your result. Draw a line to represent the expected payoff for a Hawk under different values of p. Draw another line to represent the expected payoff for a Dove under different values of p. Can you associate any meaning to the point where the two lines met? [14] Suppose the two lines met at point E in the graph above. Use the same graph to figure out if the proportion of Hawks is slightly less than E, which strategy (Hawk or Dove) has a higher payoff?
[15] If the proportion of Hawks is slightly greater than E, which strategy has a higher payoff? According to Maynard Smith (1982), this point represents an “Evolutionarily Stable Strategy”, a strategy such that, “if all the members of a population adopt it, no mutant strategy can invade”. In other words, if there are more Hawks than E more Doves survive. If there are less Hawks than E more Hawks survive. This means that the proportion of Hawks in the male population is stable at E. It is an important contribution to evolutionary biology, but a simple application of game theory.
3. Should we believe in God game?
4. Analyze the Kicker & Goalie example in the textbook (p.168) from the Goalie’s perspective, defining y=the probability of using Left strategy in the Goalie’s mix. Find the optimal y so that the success rate of the Kicker is independent of the strategy the kicker is using.
5. Analyze the “Parking meter game” (p. 164 in textbook) from the view point of the car driver. What kind of mixed strategy the car driver should use in order that he will not lose in the long run?