Scenario: Alex puts $100 on the table. Bob chooses either to take $51 for himself and $49 for Clare or pass. If Bob passes, then Alex adds another $2 and Clare chooses either to take $52 for herself and $50 for Bob or pass. If Clare passes, then Alex adds another $2 and Bob chooses either to take $53 for himself and $51 for Clare or pass, and so on. In other words, a player chooses either to take $2 more for himself/herself or pass, and, every time a player passes, Alex adds another $2. The game ends if $200 is on the table or either Bob or Clare chooses not to pass before that. If the amount on the table reached $200, then Bob and Clare get $99.99 each (Alex takes two pennies back). The figure below shows the extensive form of this game. Bob's payoff is written first and Clare's second.

-Refer to the scenario above.Suppose that,if the game reached the end with $200 on the table,Bob and Clare get $100 each instead of $99.99.Then which of the following could happen in an equilibrium?
A) Bob passes until $198 is on the table, and Clare ends the game.
B) Bob passes every time it is his turn, and Clare does the same.
C) Bob ends the game in his first turn.
D) Bob ends the game when $150 is on the table.

Question

Scenario: Alex puts $100 on the table. Bob chooses either to take $51 for himself and $49 for Clare or pass. If Bob passes, then Alex adds another $2 and Clare chooses either to take $52 for herself and $50 for Bob or pass. If Clare passes, then Alex adds another $2 and Bob chooses either to take $53 for himself and $51 for Clare or pass, and so on. In other words, a player chooses either to take $2 more for himself/herself or pass, and, every time a player passes, Alex adds another $2. The game ends if $200 is on the table or either Bob or Clare chooses not to pass before that. If the amount on the table reached $200, then Bob and Clare get $99.99 each (Alex takes two pennies back). The figure below shows the extensive form of this game. Bob's payoff is written first and Clare's second.
 
-Refer to the scenario above.Suppose that,if the game reached the end with $200 on the table,Bob and Clare get $100 each instead of $99.99.Then which of the following could happen in an equilibrium?
A) Bob passes until $198 is on the table, and Clare ends the game. 
B) Bob passes every time it is his turn, and Clare does the same. 
C) Bob ends the game in his first turn. 
D) Bob ends the game when $150 is on the table.

Quizplus · Accepted Answer

In an equilibrium, both players must be making their best choices given the actions of the other player. 
If Bob takes the $51 on his turn, Clare will be left with only $49, which is less than what she could get if she passes and waits for the $2 to be added. Therefore, Bob will pass. 
If Clare takes the $52 on her turn, Bob will be left with only $50, which is less than what he could get if he passes and waits for the $2 to be added. Therefore, Clare will pass. 
This pattern will continue until $200 is on the table, at which point both players will take the payoff of $100 each. Therefore, B is the best choice.

Scenario: Alex Puts $100 on the Table