Game Theory Review Questions

Transcription

1 Game Theory Review Questions Sérgio O. Parreiras All Rights Reserved Repeated Games What is the difference between a sequence of actions and a strategy in a twicerepeated game? Express a strategy profile as a state automata. Given a state automata find the associated value function. Given some strategy, define a one-shot deviation at some history in a repeated game. For each state check for one-shot deviations using the value function and the stage payoffs. 0.2 Cooperative Games Define a coalitional game. Define the core of a coalitional game. Describe the top-trading cycle procedure. Describe the deferred acceptance procedure. 1

2 0.3 Extensive Games Pick a game like the espionage game or the card game in the lecture notes. Describe the strategy set of each player and be sure to differentiate "actions" from "strategies". Given an extensive game, find its normal form (matrix) representation. Find Nash equilibria that fail to be subgame perfect equilibrium. 0.4 Constrained Optimization Consider the problem of maximizing log(1 + x) + 2 log(1 + y) subject to x 0, y 0, x + y 10 and x + 3y 20 and x 2 + y Write down the payoff of player 1 in the auxiliary game where player 2 chooses nonnegative penalties, λ 1, λ 2, λ 3, λ 4 and λ 5 (or subsidies) that player 1 pays (receives) if player 1 goes over (stays below) the respective constraint -- the Lagrangian of the constrained optimization problem. 2. Write down the complementary slackness conditions. 3. Write down the first-order conditions for player 1 4. List in a table all the 2 5 = 32 cases for the λs we should consider. 5. Prove in any solution we must have x > 0 and y > Continuing the previous item, what are the values of λ 1 and λ 2? 7. For each of the remaining 8 cases check if the case has a solution compatible with positive lambdas, complementary slackness solutions, the FOC and the original constrains. 0.5 Weakly and Strictly Dominated Strategies 1. Give example of a 2 by 2 matrix game where one of the players uses a weakly dominated strategy in equilibrium. 2. Consider the voting game with N > 2 voters. There are two candidates. Voters are type A or type B. Voters can vote for candidate A or vote for candidate B. A voter A prefers A's victory to win than a tie and prefers a tie than B's victory. A voter B prefers B's victory than a tie and prefers a tie than A's victory. Assume a majority of voters are A voters. Voters must cast a vote, there is no abstention Show there is a Nash equilibrium where B wins. 3. In the equilibrium above, does any player uses a weakly dominated strategy? 2

3 0.6 Mixed Strategies Pick any 3 by 3 matrix game you can find and solve for all of its Nash equilibrium (including mixed) using our step by step approach 1. For each pure strategy of player 1, write his expected utility when 2 uses a mixed strategy. 2. Derive 1's pure best response to a mixed strategy of Do steps 1 and 2 for player List the 7 cases possible for player 1: put positive prob. on all strategies, puts positive prob. on only the first and second strategy,..., puts all prob. of the last strategy. 5. Check if there is an eq. compatible with each case above (you may need to check for cases of player 2 as well). 0.7 Decision Theory: Choice Under Uncertainty Set-Up: The choice set (the set of prizes) is X = {banana, kit-kat, san pellegrino water} and δ b, δ k and δ s are the lotteries that deliver the respective prize with certainty (probability 1). Question 1 Assume Anna is expected utility maximizer and that her preferences are δ k δ s δ b and 1 2 δ k 1 2 δ b δ s. 1a In words, describe what the lottery 1 2 δ k 1 2 δ b means. 1b Which lottery she prefers δ s or 2 3 δ k 1 3 δ b? Why? Question 2 Bruce preferences are such that δ s δ b δ k. Moreover, for any two lotteries l p = p s δ s pb δ b pk δ k and l q = q s δ s qb δ b qk δ k where p s, p b, p k, q s, q b, q k 0, p s + p b + p k = 1, and q s + q b + q k = 1. Bruce chooses l p if p s > q s or also if p s = q s and p b > q b. Otherwise, he chooses l q. 2a Why the two lotteries are identical when p s = q s and p b = q b? 2b Describe Bruce's choice criterion in words only. 2c Are Bruce's preferences complete? 2d Are Bruce's preferences transitive? 2d By means of an example, show that Bruce's preferences are not continuous. 3

4 Question 3 Julia, Oscar, and Pablo utility for money are respectively: u J (x) = 3x + 1 u O (x) = ln(x) u P (x) = 100x x 2 3a Computer their respective marginal utility for money, u. 3b For each of them write their expected utility for the lottery l 1 = (( 1 4, 1 2, 1 4 ), (10, 15, 20)). (You will need a calculator) 3c Repeat the above for the lottery l 2 = (( 1 2, 1 2 ), (10, 20)). 3d For each for the decision makers, which of the two lotteries above they prefer? 3e Relate your answers in 3a and 3d. 0.8 Readings Question 4 What are the sources of risk in the natural environment of Lemurs (Madagascar)? Question 5 What are the risk sources faced by Lemurs in their natural habitat? Question 6 Summarize/explain the risk-sensitive foraging theory. Question 7 How risk-aversion is measured in the experiments described in the article Variance-sensitive choice in lemurs? Question 8 Which traits of Lemur's behavior and/or biology might be explained as adaptation to risk environment? Question 9 Accordingly to "Poor Economics" how risk affects the poor? What the poor hate (perhaps the most) about living under high risk? 0.9 Static Games Question 10 For all the games in the figure: 10a Find all the Nash Equilibria 10b Find all the efficient outcomes. 4

5 10c Plot the best response function for both players. Question 11 Pick one game in the figure figure and chose any two non-negative numbers: n 1 and n 2. Create the function f(s, t) that picks a strategy profile (s, t) and gives f(s, t) = n 1 u 1 (s, t) + n 2 u 2 (s, t) where u 1 is the payoff of player 1 and u 2 is the payoff of player 2. That is, f is some weighted sum of payoffs. 11a Find the strategy profile that maximizes f. 11b Show that the profile in 11a is efficient. Question 12 Now pick a different game from the one you selected in question 11. In this game, pick one efficient strategy profile. For this profile, due the converse of question 11: find some new non-positive weights n 1 and n 2 such that then the outcome you selected will maximize f (for the new weights that you have to find). Question 13 For the Cournot game where firms have different unit costs,u 1 (q 1, q 1 ) = (α q 1 q 2 c 1 ) q 1 and u 2 (q 1, q 2 ) = (α q 1 q 2 c 2 ) q 2. 13a Find the best response of firm 1. 13b Find the best response of firm 2. 13c Find one Nash equilibrium. 13d Assume firms merge, also assume that c 1 = c 2 find the values of q 1 and q 2 that maximize the sum of their profits (payoffs). (Hint: take the derivative of the sum of profits with respect q 1, the also take the derivative wrt. q 2, equate both equations to zero and solve for q 1 and q 2 - there are an infinite number of solutions but in all of them q 1 + q 2 is constant...) 13e How the total supply in 13d compares with the total supply in 13c? What about the price, P = α Q = α q 1 q 2? 13f Are consumers better of under 13c or 13d? Why? 5