3. DON'T MESS WITH ME GAME TREE. Suppose Player #1 (P1) anticipates the possibility of being challenged by rival Player #2 (P2). To discourage P2 from challenging P1 tomorrow, P1 could invest in mixed martial arts training today, thereby making P1 a more formidable rival in case it has to battle tomorrow. Suppose the game tree below accurately models this dynamic game. TIP: Review the dynamic games slides to refresh your memory about analyzing games trees and finding SPNE. a) b) P1 Fight (F) (100,-50) Challenge (C) MMA Training (M) Give Up (G) P2 (-40, 100) Do Not (D) (60,0) fight (f) (-20, 150) challenge (c) Pl No Training (N) give up (g) P2 (0, 100) do not (d) (100, 0) Find all Subgame Perfect Nash Equilibria (SPNE) of this game. TIP: What is the definition of SPNE? In the subgames in the game tree directly following P1's choice to invest in MMA Training, identify ALL of the noncredible threats and explain why each is noncredible.

3. DON'T MESS WITH ME GAME TREE. Suppose Player #1 (P1) anticipates the possibility of being challenged by rival Player #2 (P2). To discourage P2 from challenging P1 tomorrow, P1 could invest in mixed martial arts training today, thereby making P1 a more formidable rival in case it has to battle tomorrow. Suppose the game tree below accurately models this dynamic game. TIP: Review the dynamic games slides to refresh your memory about analyzing games trees and finding SPNE. a) b) P1 Fight (F) (100,-50) Challenge (C) MMA Training (M) Give Up (G) P2 (-40, 100) Do Not (D) (60,0) fight (f) (-20, 150) challenge (c) Pl No Training (N) give up (g) P2 (0, 100) do not (d) (100, 0) Find all Subgame Perfect Nash Equilibria (SPNE) of this game. TIP: What is the definition of SPNE? In the subgames in the game tree directly following P1's choice to invest in MMA Training, identify ALL of the noncredible threats and explain why each is noncredible.

ENGR.ECONOMIC ANALYSIS

14th Edition

ISBN:9780190931919

Author:NEWNAN

Publisher:NEWNAN

Chapter1: Making Economics Decisions

Section: Chapter Questions

Problem 1QTC

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6) You have been assigned to create a new TV game show, and you have an interesting idea that you call, “I WANT TO BE A MILLIONAIRE.” The basics are: 1) two contestants; 2) the show begins with each contestant being given $1 million (!); and then 3) they begin playing a game that can increase or decrease that $1 million. You worry that the initial outlay of $2 million will stun your producers, so you decide to prepare them with a simpler version of your game that you call: “I WANT $3.” There are four steps in this simpler game: There are two contestants/opponents (who do not know each other and cannot communicate with each other during the game). Each player is given $3 at the start of the game. Independently and simultaneously, each player must choose whether they want to add $0, $1, $2 or $3 to their initial stake of $3. Doing so reduces their opponent’s award by $0, $2, $4, or $6, respectively. Each player knows that their payoff at the end of the game is based on their initial…
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Two discount stores (Megastore and Superstore) are interested in expanding their market share through advertising. The table below depicts the strategic outcomes (profits) of both stores with and without advertising. Superstore - Advertise Superstore - Don't Advertise Megastore - Advertise $95, $80 $305, $55 Megastore - Don't Advertise $65, $285 $165, $115 When the game does reach Nash Equilibrium, the payoffs for both stores will be a) Megastore $95 and Superstore $80 b) Megastore $305 and Superstore $55 c) Megastore $65 and Superstore $285 d) Megastore $165 and Superstore $115
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Identify any strictly dominant strategies for the players.
2. Consider the following "centipede game." The game starts with player 1 choosing be- tween terminate (T) and continue (C). If player 1 chooses C, the game proceeds with player 2 choosing between terminate (t) and continue (c). The two players choose be- tween terminate and continue in turn if the other player chooses continue until the terminal nodes with (player l's payoff, player 2's payoff) are reached as shown below. TTTT Player 1 Player 2 Player 1 Player 2 (3, 3) t (1, 1) (0, 3) (2, 2) (1, 4) (a) List all possible strategies of each player. (b) Transform the game tree into a normal-form matrix representation. (c) Find all pure-strategy Nash equilibria. (d) Find the unique pure-strategy subgame-perfect equilibrium.
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3. Suppose we play the following game. I give you $100 for your initial bankroll. At each time n, you decide how much of your current wealth to bet. You cannot borrow money. You can only play with the money I gave you in the beginning or any money that you have won so far. The game is simple. At each time n ≥ 1, you decide the amount to bet. I will roll a fair die. If the die comes up 1,2,3,..., or 5, you win; if the die comes up 6, then you lose. IOW, if you bet $10 on the first roll, you will either have $90 or $110 after the first roll. (a) Suppose you wish to maximize your profit on the first roll. How much should you bet? (Most of you will get this wrong.) (b) What is the expected profit on the first roll if your bet is b with 0 ≤ b ≤ 100? (c) Suppose you wish to maximize your expected profit on the first roll. How much should you bet? (d) Suppose you wish to maximize your expected profit betting on the nth roll. How much of your current wealth do you bet? (e) Let X₂, be your…
1. Consider a game of chicken: two players approach a narrow bridge with room only for one, and each chooses either to go through and hope the other guy backs off (call this action H, for "hawk"), or backs off to let the other go through (call this action D, for "dove"). If both players are normal, then each gets a payoff of -1 from a collision (i.e. if both choose H), 1 if they choose H against D, 0 if they choose D (regardless of what the other player chooses). Now perturb the game as follows: there is a chance p > 0 that P2 is a “violent type", who gets 1 from playing H regardless of Pl's action, and -1 from playing D, again regardless of P1's action. (a) Illustrate the game tree corresponding to the Harsanyi transformation of this game (i.e. the dynamic game of imperfect information in which nature moves first). (b) Illustrate the extended game (i.e. the normal form game corresponding to your game tree in part (a)). (c) Find all values of p for which there is a BNE in which P1…