Question

In: Economics

Suppose two players take turns playing another version of the parlor game discussed in class on...

Suppose two players take turns playing another version of the parlor game discussed in class on Tuesday. In this version, players can say any whole number between 1 and 10. The first person to get the running total to 25 wins. Do you want to go first or second? Figure out the optimal strategy for this game. Show your work.

(b) (5 points) ECN-322 only for this part: Suppose two players take turns playing another version of the parlor game discussed in class on Tuesday. In this version, Player 1 can say any whole number from 1 to 5. Player 2 can say any whole number from 1 to 6. Player 1 goes first. The first person to get the running total to 25 wins. With perfect play, is Player 1 going to win, or is Player 2 going to win? Figure out the optimal strategy for the winner. Show your work.

Solutions

Expert Solution

Lets try to solve the problem from backward calculations.

A and B are 2 players.

Say that A moves first, then B moves and so on.

The person who takes the running total to 25 wins.

That means the running total before this hand should be somewhere between 15-24. (Cause they can choose between 1-10)

So, the player who takes the running total to exactly 14 will win the tournament. (because the next person should choose between 0-10, and that choice will take the total between 15-24)

Now, if A moves first and choose any number between 4-10 then A will loose. (As B can take the total to 14 easily)

So A will choose 3.

Now, lets iterate

A choose 3

B chooses any number between 1-10 (say x) (running total is between 4-13)

Depending upon the choice of B, A will choose a number (11 - x) to make the running total exactly 14

Now whatever B choose (say y), running total will be between 15-25

Depending upon the choice of B, A will choose a number (11 - y) to make the running total exactly 25

So, the one who moves first will win the game.

Player 1 : A : Can choose between 1 to 5

Player 2 : B : Can choose between 1 to 6

A moves first. Player to get running total to 25 wins.

Again, lets try to solve this in reverse direction.

A will win if running total before A's move is between 20-24.

If running total after A's move is 19 then B will win, because B can choose 6.

If running total after A's move is 18 then B will win, because B can choose 1 and running total will be 19, then A cant take it to 25, because A cannot choose 6.

So, B wins the game, if B can make the running total 19. So, before B's move the running total should be between 13-18. Therefore, B wants to take it to 12-13 and B will win the tournament.

So, B wins the game, if B can make the running total 12-13. So, before B's move the running total should be between 6-11. Therefore B wants to take it to 5 and B will win the tournament.

Suppose A will choose 1, B can choose 4 - running total 5.

Suppose A will choose 2, B can choose 3 - running total 5.

Suppose A will choose 3, B can choose 2 - running total 5.

Suppose A will choose 4, B can choose 1 - running total 5.

Running total 5, Now A's turn,

Suppose A will choose 1, B can choose 6 - running total 12.

Suppose A will choose 2, B can choose 5 - running total 12.

Suppose A will choose 3, B can choose 4 - running total 12.

Suppose A will choose 4, B can choose 3 - running total 12.

Suppose A will choose 5, B can choose 2 - running total 12.

Running total 12, Now A's turn,

Suppose A will choose 1, B can choose 6 - running total 19.

Suppose A will choose 2, B can choose 5 - running total 19.

Suppose A will choose 3, B can choose 4 - running total 19.

Suppose A will choose 4, B can choose 3 - running total 19.

Suppose A will choose 5, B can choose 2 - running total 19.

Running total 19, Now A's turn,

Suppose A will choose 1, B can choose 5 - running total 25.

Suppose A will choose 2, B can choose 4 - running total 25.

Suppose A will choose 3, B can choose 3 - running total 25.

Suppose A will choose 4, B can choose 2 - running total 25.

Suppose A will choose 5, B can choose 1 - running total 25

B wins in all above scenarios, that is if A chooses 1-4 first.

So, A will choose 5 in the first move

Even if A will choose 5, B can choose 1, running total 6

A can choose between 1-5 and can make running total 7-11.

So, B can again take running total to 12, and will win.

So, B will win in any scenario.

Rahul Sunny answered 1 year ago

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