Using induction: Show that player 1 can always win a Nim game in which the number...

Using induction: Show that player 1 can always win a Nim game in which the number of heaps with an odd number of coins is odd.

Expert Solution

Colby Messinger answered 2 years ago

How to prove that player I can always win a Nim game in which hte number...

How to prove that player I can always win a Nim game in which hte number of heaps with an odd number of coins is odd. I understand the objective, having trouble formuating a more formula argument for the problem.

The game of Nim. This is a well-known game with a number of variants. The following...

The game of Nim. This is a well-known game with a number of variants. The following variant has an interesting winning strategy. Two players alternately take marbles from a pile. In each move, a player chooses how many marbles to take. The player must take at least one but at most half of the marbles. Then the other player takes a turn. The player who takes the last marble loses. Write a program in which the computer plays against a...

Consider a game in which, simultaneously, player 1 selects any real number x and player 2...

Consider a game in which, simultaneously, player 1 selects any real number x and player 2 selects any real number y. The payoffs are given by: u1 (x, y) = 2x − x2 + 2xy u2 (x, y) = 10y − 2xy − y2. (a) Calculate and graph each player’s best-response function as a function of the opposing player’s pure strategy. (b) Find and report the Nash equilibria of the game. (c) Determine the rationalizable strategy profiles for this game.

In the game of roulette, a player can place a $ 4 bet on the number...

In the game of roulette, a player can place a $ 4 bet on the number 27 and have a StartFraction 1 Over 38 EndFraction 1 38 probability of winning. If the metal ball lands on 27, the player gets to keep the $ 4 paid to play the game and the player is awarded an additional $ 140 . Otherwise, the player is awarded nothing and the casino takes the player's $ 4 . What is the expected...

Consider a lottery game in which a person can win $0, $1, $2, or $1,000. The...

Consider a lottery game in which a person can win $0, $1, $2, or $1,000. The probability of winning nothing when one plays the game is 0.99, the probability of winning $1 is 0.009, and the probability of winning $2 is 0.0009. If the game cost $1 to play what is the probability that a person will at least win their money back in the game? What is the interval of values that are within one standard deviation of the...

Consider a game in which, simultaneously, player 1 selects a number x ∈ [0, 8] and...

Consider a game in which, simultaneously, player 1 selects a number x ∈ [0, 8] and player 2 selects a number y ∈ [0, 8]. The payoffs are given by: u1(x, y) = 2xy − 8x − x 2 u2(x, y) = 4xy − y 2 (a) Calculate and graph the best responses for each player. (b) What strategies are never played? (c) Find all Nash equilibriums? (d) What is the preferred equilibrium in society? (e) How would we classify...

In the game of roulette, a player can place a $4 bet on the number 14...

In the game of roulette, a player can place a $4 bet on the number 14 and have a StartFraction 1 Over 38 EndFraction probability of winning. If the metal ball lands on 14, the player gets to keep the $4 paid to play the game and the player is awarded an additional $140. Otherwise, the player is awarded nothing and the casino takes the player's $4. What is the expected value of the game to the player? If...

In the game of roulette, a player can place a $9 bet on the number 6...

In the game of roulette, a player can place a $9 bet on the number 6 and have a 1 /38 probability of winning. If the metal ball lands on 6, the player gets to keep the $9 paid to play the game and the player is awarded an additional $315 Otherwise, the player is awarded nothing and the casino takes the player's $9. Find the expected value E(x) to the player for one play of the game. If x...

Consider the following two-player game, in which Player 1 is the IMF, and Player 2 is...

Consider the following two-player game, in which Player 1 is the IMF, and Player 2 is a debtor country. Reform Waste Aid 3, 2 -2, 3 No Aid -2, 1 0, 0 a) Compute all (pure and mixed) Nash equilibria. b) Do you think that the above game is the case of a resource curse? Interpret the game with a story of a resource curse.

There are two players in the game. Each player can pick any integer number between 1...

There are two players in the game. Each player can pick any integer number between 1 and n. If two numbers are the same then player 1 pays 1 dollar to player 2. If two numbers are different than nothing happens. (a) Prove that there are no equilibria in pure strategies; (b) Prove that in the equilibrium each strategy should be played with a positive probability. (c) Find all NE of the game.

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