There are two players, each holding a box. At the beginning of the game, each box...

There are two players, each holding a box. At the beginning of the game, each box contains one dollar. Player 1 is offered the choice between stopping the game and continuing. If he chooses to stop,then each player receives the money in his own box and the game ends.If Player 1 chooses to continue, then a dollar is removed from his box and two dollars are added to Player 2’s box. Then Player 2 must choose between stopping the game and continuing. If he stops, then the game ends and each player keeps the money in his own box. If player 2 continues,then a dollar is removed from his box and two dollars are added to Player 1’s box. Play continues like this, alternating between the players,until either one of them decides to stop or 5 rounds of play have elapsed (i.e., until Player 1 chooses for the third time). If neither player chooses to stop by the end of the 5th round, then both players get zero.

In the backward induction outcome of this game Player 1 obtains a payoff of__ and Player 2 obtains a payoff of __ (Please, enter only numerical values like 0, 1, 2, ...)

Expert Solution

Solution. Since, This is the problem of standard centipede game of sequencial games and its game tree

is showing by image below.

Game is playing for five periods so our histories will be denoted as the h(t) where t stands for time period 1, 2, 3, 4, 5 and they are:

h(5)= (CCCC) ; h(4)= (CCC) ; h(3)= (CC), h(2)= (C) ; h(1)= where history at period 1 is null.

Using the Backward Induaction techniques we can easily trace using game tree that at history period 5, player 1 plays game by chossing S or C.

Player 1: S= $3 , C = $0; so playing S is the best strategy for him.

Since, Player 1 is going to play S in period 5 player 2 anticipating that player 1 is a rational he makes his decision to choose S or C at period 4.

At period 4, Player 2: S=$4 , C= $3; so playing S is the best strategy for him.

Similarily for player 1, 2 and 1 playing S is the best strategy in period 3, 2 and 1 respectively.

So strategy (SSSSS) is the subgame perfect equilibrium for this game.

Player 1 and 2 will get ($1, $1) as a final outcome where player 1 will choose not to play in period 1.

Rahul Sunny answered 2 years ago

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