Consider the following simple model of the Mile End Park Beautification Initiative (from the lectures). There are two students, each of whom can choose to plant flowers (F) or not to plant flowers (N). If both students plant flowers, then the appreciation) benefit is 95 to each player. If exactly one student plants flowers, then the benefit is 50 to each player. The cost to either pla planting flowers is c > 0, regardless of the other player's choice. Each student starts (before any costs or benefits) with a payoff of zero. The above assumptions lead to the following game in strategic form. F 95 — с, 95 — с 50 - с, 50 N 50, 50 - с 0,0 Suppose thatc< 95.
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- THE PRISONERS' DILEMMA: ESSENTIAL FEATURES Consider the following simple model of the Mile End Park Beautification Initiative (from the lectures). There are two students, each of whom can choose to plant flowers (F) or not to plant flowers (N). If both students plant flowers, then the (flower appreciation) benefit is 130 to each player. If exactly one student plants flowers, then the benefit is 60 to each player. The cost to either player of planting flowers is c> 0, regardless of the other player's choice. Each student starts (before any costs or benefits) with a payoff of zero. The above assumptions lead to the following game in strategic form. 130- с, 130с 60 - с, 60 60, 60 c 0,0 Suppose that e< 130. This game is a prisoners' dilemma if and only if the parameter c is strictly greater than which value? Answer:Consider the following game - one card is dealt to player 1 ( the sender) from a standard deck of playing cards. The card may either be red (heart or diamond) or black (spades or clubs). Player 1 observes her card, but player 2 (the receiver) does not - Player 1 decides to Play (P) or Not Play (N). If player 1 chooses not to play, then the game ends and the player receives -1 and player 2 receives 1. - If player 1 chooses to play, then player 2 observes this decision (but not the card) and chooses to Continue (C) or Quit (Q). If player 2 chooses Q, player 1 earns a payoff of 1 and player 2 a payoff of -1 regardless of player 1's card - If player 2 chooses continue, player 1 reveals her card. If the card is red, player 1 receives a payoff of 3 and player 2 a payoff of -3. If the card is black, player 1 receives a payoff of 2 and player 2 a payoff of -1 a. Draw the extensive form game b. Draw the Bayesian form gameIn Las Vegas, roulette is played on a wheel with 38 slots, of which 18 are black, 18 are red, and 2 are green (zero and double-zero). Your friend impulsively takes all $361 out of his pocket and bets it on black, which pays 1 for 1. This means that if the ball lands on one of the 18 black slots, he ends up with $722, and if it doesn't, he ends up with nothing. Once the croupier releases the ball, your friend panics; it turns out that the $361 he bet was literally all the money he has. While he is risk-averse - his utility function is u(x) =, where x is his roulette payoff - you are effectively risk neutral over such small stakes. a. When the ball is still spinning, what is the expected profit for the casino? b. When the ball is still spinning, what is the expected value of your friend's wealth? c. When the ball is still spinning, what is your friend's certainty equivalent (i.e., how much money would he accept with certainty to walk away from his bet). Say you propose the following…
- Question 1 Consider the following game. Player 1 has 3 actions (Top, middle,Bottom) and player 2 has three actions (Left, Middle, Right). Each player chooses their action simultaneously. The game is played only once. The first element of the payoff vector is player 1’s payoff. Note that one of the payoffs to player 2 has been omitted (denoted by x). A) Suppose that the value of x is such that player 2 has a strictly dominant strategy. Find the solution to the game. What solution concept did you use to solve the game? B) Suppose that the value of x is such the player 2 does NOT have a strictly dominant strategy. Find the solution to the game. What solution concept did you use to solve the game?Imagine a game where individuals can be either cooperative (like splitting a resource) or selfish (like grabbing the entire resource). Depending on the relative costs and benefits of interacting and the resource, there might be a variety of possible payoff matrices for such an interaction. Of the following matrices, which one illustrates the largest “temptation to cheat?”The US and Canada have overfished North Atlantic cod stocks nearly to the point of extinction. Both countries wish to preserve the cod industry (and hence the cod stocks), so the two countries sign an agreement to limit fish hauls. Each country has two possible strategies: comply with the agreement (limit fishing) or renege on the agreement (overfish). Each country’s payoffs for different strategy combinations are given in the matrix below. The numbers in the cells represent utilities, and the payoff ordering is (US, Canada). QUESTIONS: 1.If this is a one-shot game (i.e., it is played once), do the players have a dominant strategy? If so, what is it? Briefly explain your answer. 2.What is the equilibrium of this game? Is it Pareto-optimal? How do you know?
- In the sequential production game shown below, the actions of four players are required in order for a company's project to be successful (for example, each player could be a different division or department with the company). Player 1 Player 2 Player 3 Player 4 Act Act Act Act 10 10 10 Quit Quit Quit Quit 10 -X -X -X -X -X -X Suppose x = 5,000 (payoffs are -5,000). Suppose player 1 believes that Player 4 will quit with probability p. How does this change the game? %3D O a) If 0 0.002, Player 1 will quit. O c) If p > 0, Player 1 will quit d) The value of p is irrelevant to Player 1 O o o oAlice and Julia want to spend time together but have different preferences for their activities. They can choose between two options: going to a soccer game or watching a football game. Alice prefers football and Julia prefers the football game, but they both prefer being together over being apart. If they both choose to attend a soccer game then Julia receives a payoff of 2 and Alice a payoff of 1. If they both choose to attend a football game, then Julia receive a payoff of 1 and Alice a payoff of 2. If one chooses a football game and the other a soccer game then they both receive a payoff of O. The payout matrix below illustrates these payoffs. Which of the following statements are true: I. There is a dominant strategy II. A nash equilibrium exists where Alice attends a football game and Julia attends a soccer game III. A nash equilibrium exists where both friends attend a soccer game IV. Both attending a football game is a stable outcome. Alice Soccer Football Julia Soccer Football…UNIT 9 CHAPTER 5 In a gambling game, Player A and Player B both have a $1 and a $5 bill. Each player selects one of the bills without the other player knowing the bill selected. Simultaneously they both reveal the bills selected. If the bills do not match, Player A wins Player B's bill. If the bills match, Player B wins Player A's bill. Develop the game theory table for this game. The values should be expressed as the gains (or losses) for Player A. Is there a pure strategy? Why or why not? Determine the optimal strategies and the value of this game. Does the game favor one player over the other? Suppose Player B decides to deviate from the optimal strategy and begins playing each bill 50% of the time. What should Player A do to improve Player A’s winnings? Comment on why it is important to follow an optimal game theory strategy.
- Write out the extensive form of a game between a farmer (playing in the first round) and nature (playing a mixed strategy in the second round). Assume that the farmer can either pay a cash rent of $1500 for land (English system) or 1/2 of crop production to the landlord (sharecropping). Assume the farmer is planting corn and will produce 2 tons of corn. Assume that nature has a 50% chance of playing a strategy in which the price of corn is $3500/ton and a 50% chance of playing a strategy in which the price of corn is $500/ton. The farmer keeps any money left after paying cash rent and sells any corn left after paying the landlord in sharecropping. What is the most that a risk neutral farmer would be willing to pay for an accurate prediction of the price of corn in problem 1 before choosing whether to pay cash rent or sharecrop?Question 1 Consider the following game. Player 1 has 3 actions (Top, middle,Bottom) and player 2 has three actions (Left, Middle, Right). Each player chooses their action simultaneously. The game is played only once. The first element of the payoff vector is player 1’s payoff. Note that one of the payoffs to player 2 has been omitted (denoted by x). 1. Determine the range of values for x such that Player 2 has a strictly dominant strategy.Hello, please help me to solve part (d) and (e):Charlie finds two fifty-pence pieces on the floor. His friend Dylan is standing next to him when he finds them. Chris can offer Dylan nothing at all, one of the fifty-pence pieces, or both. Dylan observes the offer made by Charlie, and can either accept the offer (in which case they each receive the split specified by Charlie) or reject the offer.If he rejects the offer, each player gets nothing at all (because Charlie is embarassed and throws the moneyaway).(a) Formulate this interaction as an extensive-form game. To keep things simple, players’ payoff is equal to their monetary gain.(b) List all histories of the game. Split these into terminal and non-terminal histories.(c) What are the strategies available to Charlie? What are the strategies available to Dylan? Draw the strategic-form game.(d) Find the pure-strategy Nash equilibria of this game.(e) What do you think will happen?