MIDTERM REVIEW QUESTIONS The Midterm Exam will take place in class on Friday, Feb. 6. Please bring one or more blank blue books and a pen to the exam. Please answer all questions completely, and, where applicable, show your work. Answers without any indication of how you calculated them earn zero credit. NOTE: In many of the problems below, I make the same STANDARD ASSUMPTION, which is: THE PLAYERS ARE INDIVIDUALISTICALLY RATIONAL AGENTS WITH COMPLETE (AND, IF RELEVANT, PERFECT) INFORMATION. 1. Define or explain the following terms, being sure to distinguish between the items in a list of items. Feel free to use examples in your explanations. certainty/risk/uncertainty utility/expected utility Pareto inferior/Pareto superior/Pareto optimal extensive form/normal form of a game purely competitive/purely cooperative/mixed motive game parametric/strategic rationality complete/perfect information pure/mixed strategy information set maximin/minimax pure strategy saddle point strict/weak dominance equalizer mixed strategy Nash equilibrium equivalent strategy combinations interchangeable strategy combinations strong/weak sense of "collective action problem" supergame/subgame TwoPerson Prisoner's Dilemma Supergame CTFT/DTFT/C/D/ conditional cooperation/unconditional cooperation in PD Supergame collectively stable strategy (stable against invasions by individuals and by clusters) NPerson Prisoner's Dilemma (or Multiperson Prisoner's Dilemma) NPerson Prisoner's Dilemma Supergame Individualistic Rationality (IR) Rational SelfInterest Theory (RST) Methodological Individualism (MI) in the social sciences 2. In the sense in which we use the term, what is a game? 3. In RockScissorsPaper, two players must simultaneously choose either Rock (R) or Scissors (S) or Paper (P). Rock beats Scissors, Scissors beats Paper, and Paper beats Rock; any other combinations are ties. (a) Represent an outcome in which one player beats the other as having utility of 1 for the winner and utility of 1 for the loser; represent a tie as utility of 0 to each player. Draw a decision matrix with the three strategies (R, S, and P) for each player. Circle all pure strategy Nash Equilibrium combinations. Explain how you found them. If there are none, explain how you determined that there are none. (b) There is a mixed strategy Nash equilibrium combination. It is the same for both players: {<1/3, R>, <1/3, S>, <1/3, P>}. If one player plays this mixed strategy, what is the expected utility of the other player's pure strategies against it. Show your work. (One calculation is sufficient.) (c) Suppose one player did not play his/her Nash equilibrium mixed strategy. Suppose that particular player liked S more than the other alternative choices, so s/he adopted the following mixed strategy: {<1/4, R>, <1/2, S>, <1/4 P>}. Call this strategy the Sbiased strategy. What would be the best reply to the Sbiased strategy? If one player played the Sbiased strategy and one player played the best response to it, what would be the expected utility for each player? 4. On the STANDARD ASSUMPTION, find all IR solutions to the following twoperson, zerosum game:

























5. (a) Explain why all possible outcomes of a twoperson, zerosum game are Pareto optimal. (b) Explain why no zerosum game can be a strong or weak CAP. 6. (a) The eponymous John Nash proved that all games have at least one Nash equilibrium combination of strategies. What is the maximum number of pure strategy Nash equilibrium combinations that there can be in a simple 2x2 game, if no two of the equilibrium combinations are equivalent? Give an example. (b) All twoperson zero sum games have an IR solution. Why don't all games have an IR solution? Use an example to explain your answer. 7. Find all pure and mixed Nash equilibrium combinations of strategies in the following two person games: (a) Standard Convention Game (b) Pareto Convention Game (c) Chicken Game (d) Battle of the Sexes/Hero. 8. Axelrod identifies four characteristics of CTitforTat that he uses to explain its success in his computer tournaments. Identify and explain each of the four characteristics, and explain how each of them contributed to CTitforTat's success in Axelrod's tournaments. 9. What is the Rendezvous Game? Identify two kinds of factors other than purely IR factors can help to produce a "solution" to a Rendezvous Game. Give an example of each. 10. Assume that the LiveandLetLive System in the trenches in World War I was an example of conditional cooperation (typically CTitforTwoTats) on the part of the two armies. (Ignore the issue of how individual soldiers were motivated to do their part in the conditionally cooperative strategies of their respective armies. Just assume that they were rationally motivated to do so.) What assumptions are necessary to show that the situation of the two armies was a 2Person PD Supergame and that a combination of conditionally cooperative strategies by the two armies was a Nash equilibrium combination? Was each of the necessary assumptions satisfied in the trench warfare in which the LiveandLetLive System arose? If any of the assumptions were not satisfied, was it well enough approximated to make it seem plausible that the combination of conditionally cooperative strategies a Nash equilibrium combination? Explain your answers. Use the information provided by Axelrod and your best guesses about other relevant facts in your discussion of each assumption. 11. Critically evaluate Taylor's claim on p. 78 [Reader, p. 24] that all other equilibria of the PD Supergame are either equivalent to or Paretodominated by at least one of those that he considers. 12. For each of the following twoperson games, draw the Schelling diagram of an nperson analogue, and explain why it is analogous to the twoperson game: (a) Prisoner's Dilemma (b) Standard Convention Game (c) Pareto Convention Game (d) Chicken Game 13. In a section of his book that we did not read, Taylor says that there are Chickens nesting in the NPerson Prisoners' Dilemma supergame. To see what he means, consider an Nperson PD supergame, and consider the two strategies D and B_{m} (where m < N1). Draw a Schelling diagram for the two strategies, with L = D and R = B_{m}. (It is not necessary to calculate numerical utilities for the different combinations of strategies, the shape of the diagram is what is important, particularly at the point at which there are m other agents choosing B_{m}. Is the matrix an Nperson Chicken Game? Explain your answer. 14. "Expressway traffic is flowing toward the city during morning rush hour. Suddenly, brake lights flash, cars slow down, traffic crawls. An accident has occurred in one of the outbound lanes. Why is it the citybound traffic jams up? Drivers have reduced their speed to get a glimpse of the wreckage on the other side of the divider. Eventually, large numbers of commuters spend an extra ten minutes driving, for a tensecond look. It costs each driver ten minutes to get his look. But he pays ten seconds for his own look and nine minutes, fifty seconds for the curiosity of the drivers ahead of him. It is a bad bargain." [from the backcover of Schelling's Micromotives and Macrobehavior]. (a) David is one commuter who has just, at this moment, reached the scene of the accident. He must decide in the next three seconds whether or not to slow down and look. Is he, at that precise moment, in an MPD [MultiPerson Prisoners' Dilemma] with the other commuters? Explain your answer in a way that shows you are familiar with Schelling's definition of an MPD. [Hint: David has already been delayed 9 minutes and 50 seconds by the commuters in front of him who stopped to look.] (b) Suppose there available at no charge an audio tape that commuters can use while they sleep to eliminate the urge to slow down and look at a traffic accident. In such circumstances, the urge to slow down and look is replaced by a compulsion to keep their eyes on the road straight ahead and not to slow down. The tape works after only one night of use, and has no adverse side effects. If all commuters use the tapes, then there will be no more delays due to drivers slowing down to look at an accident scene. All commuters, including John, have received a copy of the tape in the mail. Each is trying to decide whether or not to use the tape. Assume that all the commuters would prefer to look at the traffic than not, if the cost were only a 10 second delay in their commute. Thus, if there were no other commuters, each commuter would not use the tape. But all commuters would be willing to forego looking at an accident, if by doing so they could save 10 minutes of commute time. Thus, if using the tape would save them 10 minutes of commute time in such circumstances, they would all use the tape. Explain how the decision situation of John and the other commuters might well be an MPD. Draw a Schelling diagram which illustrates how the situation could satisfy the four conditions of an MPD. 15. What is the "Folk Theorem" of NonCooperative Game theory? Illustrate it with an example. 16. This question concerns the example of the Potato Farmer (PF) and the Broccoli Farmer (BF) discussed in class. Take the STANDARD ASSUMPTION as given. Also suppose that the situation is one in which the crops are perishable, so they must be consumed when they come in, and that each year the potato crop comes in well before the broccoli crop (which, in turn, comes in well before the next year's potato crop). Thus, PF must decide whether to share some of this year's potatoes with BF before BF must decide whether to share some of this year's broccoli with PF (and BF will have to decide whether to share this year's broccoli with PF well before PF must decide whether to share some of next year's potato crop with BF). (a) OneShot Interaction. Assume only a onetime interaction (each player has an opportunity to share once), with utilities given by the matrix below. (Perhaps one farmer plans to move away next year.) Explain why the game is a TwoPerson Prisoner's Dilemma. PLAYER 2 (Broccoli Farmer)












(b) FixedLength Supergame. Suppose that BF plans to move away in four years, but that they both expect to have four years of potential sharing. Use backward induction to explain why IR agents would not share, even though they could potentially obtain the benefits of four years of trading. (c) Supergame of Indefinite Length. Suppose that PF and BF have the potential for sharing for the indefinite future (e.g., neither plans to move away and both are young and in good health). Suppose that the longterm interaction satisfies the conditions for a PD Supergame with discount factor = .99 for each agent. (i) Prepare a supergame matrix with the following strategies for each agent: D; CTFT; and DTFT. (ii) Is the combination (B,B) a Nash Equilibrium in the matrix in (i) above? Explain. (iii) Is there an IR solution to the matrix in (i) above? Explain. (iv) Is the combination (B,B) a Nash equilibrium in the full supergame (not limited to the matrix in (i) above, which shows only three of the infinitely many possible supergame strategies)? Explain. (v) Is there an IR solution to this supergame? Explain. (vi) What additional factors might produce a solution to this supergame? Explain.