Fall 2012 Game Theory In The Social Sciences Problem Set 2 D

Fall 2012game Theory In The Social Sciences Problem Set 2due In Lect

In lecture we studied how political parties or candidates choose their positions when all they care about is winning. This question explores what happens when political candidates care not only about winning but also about the policies they espouse. The U.S. Congress is up for grabs in the election this November.

Bipartisanship has again broken down, and the parties are very polarized. The more liberal faction of the Democratic Party now dominates that party and a more conservation faction dominates the Republican Party. As the election approaches, these parties are trying to stake out positions that reflect their own policy preferences and will attract enough voters to win. To simplify matters, suppose the parties have to choose a position along a left-right spectrum and can adopt one of the following positions: Liberal (L), Liberal leaning centrist (LC), Middle of the Road (M), Conservative Centrist, (CC), Conservative (C). These positions are represented on the line below where the distance between any two neighboring positions is the same.

Since the voter’s ideal points are evenly distributed along the political spectrum, the party whose position is closer to the middle of the road (M) wins. If, for example, the Republicans, R, choose a position of CC and the Democrats, D, announces, L, then R would win because CC is closer to M. If the parties run on platforms that are equally distant from M, each party is equally likely to win. Finally, each party chooses its platform secretly. As noted above, R and D care about policies as well as winning.

D’s von Neumann-Morgenstern payoffs are: 5 for winning with L; 4 for winning with LC; 3 for winning with M; 2 for winning with CC; 1 for winning with C; -1 for losing with L; -2 for losing with LC; -3 for losing with M; -4 for losing with CC; and –5 for losing with C. R’s von Neumann-Morgenstern payoffs are the opposite: 5 for winning with C; 4 for winning with CC; 3 for winning with M; 2 for winning with LC; 1 for winning with L; -1 for losing with C; -2 for losing with CC; -3 for losing with M; -4 for losing with LC; and –5 for losing with L.

(a) Suppose R adopts position C and D chooses LC. What are their payoffs?

(b) Suppose R adopts C and D chooses the more extreme position L. What are the parities payoffs?

(c) Specify the strategic form of this game.

(d) Solve the game by iterated deletion of dominated strategies. Be sure to indicate the order of deletion, what dominates what, and whether this is strict or weak dominance.

(e) What is/are the pure strategy Nash equilibria of this game?

(f) Finally, suppose that there are three parties instead of two. Call the third party S (for spoiler) and assume that S has the same preferences that D does. Consider the three-player game in which each player selects his position secretly; each voter votes for the party whose position is closest to her ideal point; and the party that receives the most votes wins. If each candidate chooses M, is this a Nash equilibrium of the game? Explain why or why not.

2. In class and in the problem above, we have seen that even ideologically motivated parties (i.e., parties that care about the policies they implement as well as being elected) will still be drawn to the center of the political spectrum and run on policy platforms that the median voter favors. This question examines what happens when the parties care about policies and there is uncertainty about what policies the median voter prefers. Two parties, Dem and Rep, simultaneously have to choose one of three party platforms Left (L), Center (C), and Right (R) as illustrated in the figure. If Dem wins with policy L, its payoff is 4+p where p measures how much it cares about policy and not just winning. If Dem wins with policies C or R, its payoff is 4. And Dem’s payoff to losing is 0. If Rep wins with policy R, its payoff is 4+p where, again, p measures how much it cares about policy and not just winning. If Rep wins with policies C or L, its payoff is 4. And Rep’s payoff to losing is 0. The party that runs on a policy closer to the median voter’s preference wins. If, for example, the median voter prefers L and Dem and Rep run on platforms C and R, respectively, then Dem wins (because C is closer to L) with payoffs 4 and 0 for Dem and Rep. If the parties’ platforms are equally far from the median’s preferred policy, each party wins with probability ½ and loses with probability ½. The figure shows the preferred policies L, C, and R.

However, the parties are unsure of the median voter’s preference. Polling data indicates that there is a 25% chance that the median favors L, a 50% chance that the median favors C, and a 25% chance that the median favors R.

The strategic form for this game is: Rep L C R L 2 , 2 2 p — 3 , 4 1 p — ?, ? Dem C 3, , ,3 p — R 2, , ,2 p —

(a) Will an ideologically motivated party ever choose the extreme position favored by the other party. More concretely, is it ever rational for Dem to choose R? Explain why or why not. In answering (b)- (d) assume that the premium to winning with one’s preferred policy is p=6, so that Dem’s and Rep’s respective payoffs to winning with L and R are 4+p=10.

(b) Suppose Dem chooses L and Rep chooses R. The payoffs to these actions depend on the median’s preferred policy. Suppose the median prefers L, what are Dem’s and Rep’s payoffs? What are Dem’s and Rep’s expected payoffs if the median’s preferred policy is C?

(c) Recall that the parties are unsure of the median voter’s preference with a 25% chance that the median favors L, a 50% chance that the median favors C, and a 25% chance that the median favors R. What is Dem’s expected payoff to the strategies (L,R)? What is Rep’s expected payoff?

(d) What is/are the Nash equilibria of the game?

(e) If the parties do not care enough about policy (i.e., if p is small enough), they will be drawn to the center. For what values of p is (C,C) the solution when the game is solved by iterated deletion of strictly dominated strategies?

In international relations, states are sometimes assumed to be concerned about how well they are doing relative to other states. This problem examines that issue. Recall the divide the dollar game discussed in class. There are two players, 1 and 2, and each secretly decides how much of the dollar to demand. If the sum of the demands is less than a dollar, each player receives what it demanded. If the demands exceed a dollar, each receives nothing. In class we also assumed that each player only cared about its monetary payoff and showed that any division of the dollar was a Nash-equilibrium outcome. (a) Verify that .25 for player 1 and .75 for 2 is a Nash-equilibrium outcome by specifying a strategy for each player that produces this outcome and by showing that neither player can benefit by deviating from its strategy given that the other player follows its strategy. Now assume that each player cares in part about how well it does relative to the other player. Suppose in particular that if 1 receives a monetary payoff of m1 and 2 receives a monetary payoff of m2, then 1’s utility to this outcome is u m m ï€ ( ). (Note that 1’s utility increases as its monetary payoff, m1, increases and decreases as the difference between 2’s payoff and its own increases. This last part formalizes the assumption that 1 cares about how well it does compared to 2.) 2’s utility is given by u m m  ï€ ( ). (b) Is .25 for 1 and .75 for 2 a Nash equilibrium outcome. Be sure to justify your answer. (c) What set of divisions can be rationalized as Nash-equilibrium outcomes given that the players care about their relative gains?

4. In this problem you will trace out a person’s von Neumann-Morgenstern utility function for money. Let L(p) be the lottery that pays $1000 with probability p and zero with probability 1-p. To anchor the scale, take u(0) = 0 and u(1000) = 1. We begin by telling our subject that p = 0.2 and asking how much money we would have to give her in order to make her indifferent between that amount of money and the lottery when p = 0.2. She answers 8 dollars. We then repeat the question for different values of p. The table summarizes the results: p 0.0 0.2 0.4 0.6 0.8 1.0, and the corresponding certainty equivalents are $0, $8, $64, $216, $512, $1000.

(a) Draw this person’s von Neumann-Morgenstern utility function.

(b) Using the graph in (a), consider the lottery which pays $4 with probability ¼ and $10 with probability ¾. What is the expected utility of this lottery? What is the certainty equivalent of this lottery?

(c) Consider the lottery that pays zero with probability 1/8, $2 with probability 1/3, $6 with probability ¼, and $10 with probability 7/24. What is the utility of this lottery?

5. Many states, including California, now use lotteries as a way of raising money. Are people who purchase lottery tickets risk acceptant, risk neutral, or risk averse? Be sure to explain your answer.

6. Consider the inspection game below where the numbers in the cells are the player’s monetary payoffs. Inspector Confess Not Confess, Conspire 9, 25 0, –,- 7. Following the events of September 11, the United States increased its airport security. The inspection game below studies several aspects of the problem. In the game, a Challenger has to decide whether or not to challenge security and Security has to decide whether or not to inspect. (Assume further that all payoffs are Von Neumann-Morgenstern.) The Challenger’s payoff to challenging and being inspected, t, depends on how good security is. Suppose that an effort to challenge security will be detected with probability d. If the challenge is detected, the Challenger’s payoff to being caught is –8. If the challenge is not detected, the Challenger’s payoff is the same as it is if there is no inspection, i.e., the Challenger’s payoff is a. Challenger Challenge Security (c) Not Challenge Inspect (i) (d) -5, t 0, 5 Security Not Inspect -15, a 5, 0 In parts (a)-(e), assume that the payoff to a successful challenge is 12, i.e., a=12. (a) What is the Challenger’s payoff t to attempting to breach security if security is initially not very good and the probability of detection is only ¼, i.e., if d=¼? (b) What is/are the equilibria of the game given that the probability of detection is only ¼? (c) For (c)-(g), suppose security improves and the probability of being detected rises to ¾. (c) What is the Challenger’s payoff t to attempting to breach security? (d) Draw Security’s best-reply correspondence? (e) What is the equilibrium probability that Security inspects? That the Challenger challenges? (f) Suppose that the Challenger becomes more determined. In particular, the payoff to a successful challenge, a, increases from 12 to 20. What is the equilibrium probability that Security inspects? (g) Although the Challenger becomes more determined, the probability of a challenge is the same in (e) where a=12 and in (f) where a=20. Why doesn’t this change in the Challenger’s payoffs affect its strategy? (h) President Obama signed the Dodd-Frank Wall Street Reform and Consumer Protection Act into law in July 2010. This act creates a Consumer Financial Protection Bureau which will try to ensure that financial services companies follow standard practices in making loans. This problem discusses the design of an auditing program for a financial services company, F. There are two options. The first is a program that is expensive to set up but has a low cost-per-audit. The second is less expensive to set up but has a high cost-per-audit. In the game below, B must decide whether or not to audit F. F must decide whether to comply with standards or evade. If F complies, it gets profits of 40. If it evades and is not caught, it gets 80; if caught, it pays 120 penalty. B pays a set-up cost s and has a per-audit cost c. With c=20 and s=1000, find B’s best reply and the mixed equilibrium. Then with c=60 and s=500, do the same. Which setup results in higher compliance? How do changes in s influence strategies? (a) Draw B’s best reply. (b) Find mixed equilibrium for c=20, s=1000. (c) For c=60, s=500, find equilibrium. (d) Which setup results in higher compliance? (e) How do s changes affect strategies? (f) Discuss the implications of high setup costs versus high per-audit costs. (g) Discuss policy considerations based on this model.

Paper For Above instruction

Politics and strategic positioning are central themes in understanding electoral competition and policy choice. The models studied in game theory reveal how parties, motivated by both winning and policy preferences, select platforms to maximize their payoff. When parties care about policies, their strategic behavior becomes more complex, involving considerations of voter distribution, policy proximity, and the strategic incentives created by uncertainty about voter preferences.

Question 1: Strategic Positioning in Two-Party Competition

The initial setup involves two parties choosing positions along a political spectrum with five possible positions: Liberal (L), Liberal leaning centrist (LC), Middle of the Road (M), Conservative Centrist (CC), and Conservative (C). The voter distribution is assumed uniform, and the candidate that is closer to the median voter — represented by the position M — wins. Both parties select their platforms secretly, caring about both winning and policy preferences, which influences their strategic choices.

Payoffs vary depending on whether a party wins or loses and the specific policy platform it adopts, with a detailed payoff structure based on proximity to voter preferences. For instance, if R adopts C and D chooses LC, their payoffs are computed based on their proximity to the median. These strategic interactions are modeled as a game with incomplete information, where each party's payoff depends on strategies chosen secretly by both sides.

Using dominance and iterated elimination, we analyze the strategic form to identify the pure strategy Nash equilibria. Through the iterative process, dominated strategies are removed, leading to equilibrium conclusions about the parties' strategic choices. This approach reveals how equilibrium selection hinges on preferences, payoff structures, and the strategic environment.

When extending the model to three parties with a spoiler S, the analysis becomes more complex, but the core principles about strategic positioning, equilibrium stability, and voter behavior remain pivotal. Questions about the stability of centered strategies versus extreme platforms are examined to understand how parties might behave under various electoral uncertainties.

Question 2: Policy Candidacy Under Uncertainty

The model examined considers the strategic trade-offs faced by parties about policy platforms under uncertainty about voter preferences. When parties care about policies with a weight p, they may choose extreme positions if the payoff justifies it. However, given the uncertainty, a careful expected payoff analysis shows that centrist strategies often prevail, especially when policy concern is moderate or voter distribution is uniform.

The example where parties choose platforms L, C, or R, with uncertain voter preference distribution, reveals the delicate balance between policy commitment and strategic positioning. Calculations of expected payoffs under different median preferences help illustrate how parties might coordinate or differentiate their platforms, leading to various Nash equilibria, including pure and mixed strategies. When the policy importance parameter p is small, centrist strategies like (C,C) dominate due to their higher expected payoffs under voter uncertainty.

International and Relative Payoff Games

In international relations, states’ concern for relative standing can be modeled via divided dollar or contest games. One such model involves two players demanding shares of a dollar, with total demands exceeding or falling short of a dollar. The strategic equilibrium involves each player balancing their demands against the risk of receiving nothing, with risk attitudes influencing strategic choices.

When players care about relative gains, utility functions incorporate differences in payoffs, leading to modified equilibrium strategies. For instance, when players maximize relative utility, pure and mixed equilibria shift, emphasizing the importance of strategic considerations like incentives for over-demanding or mutual restraint, depending on risk attitudes.

Risk and Utility in Lottery and Payment Games

The von Neumann-Morgenstern utility function is graphically constructed from uncertain payoffs, illustrating risk preferences. The curvature of this utility function reveals whether individuals are risk averse, risk neutral, or risk accepting. Calculations of expected utility under various lotteries further quantify these preferences, emphasizing the role of risk attitudes—important for understanding behavior in lotteries, financial decisions, and policy implications.

Regarding lotteries, human behavior often exhibits risk aversion, especially in context of gambling or state-sponsored lotteries, where participants tend to overvalue certainty or undervalue high-variance lotteries.

Security and Enforcement Games

Inspection and security games model strategic interactions in contexts like airport security or cybersecurity. The equilibrium analysis shows how probability of inspection or challenge affects the strategies of agents and the likelihood of detection. Variations in parameters, such as the cost of detection or the benefit of challenging, influence the equilibrium probabilities of inspection and offending.

When security improves (higher detection probabilities), the