Imagine a two player game that proceeds as follows a pot of...

Imagine a two-player game that proceeds as follows. A pot of money is created with $6 initially. Player 1 (Alice) moves first, then player 2 (Bob), then player 1 again, and finally player 2 again. At each player's turn to move, he or she has two possible actions: grab (G) or share (S). If the player grabs, they get 2/3 of the current pot of money, the other player gets 1/3 of the pot, and the game ends. If they share, then the size of the current pot is multiplied by 3/2, and the next player gets to move. At the last stage in which Bob moves, if he chooses to share, then the pot is still multiplied by 3/2, and Bob gets 1/3 of the pot, and Alice gets 2/3 of the pot. a. Model this as an extensive-form game and draw the game tree. Is it a game of perfect or imperfect information? b. How many terminal nodes does the game tree have? How many information sets are there? What about the number of subgames? c. Please use backward induction to find the subgame-perfect Nash equilibrium.

Transcript

00:01 Imagine a two -player game that proceeds as follows.

00:04 A pot of money is created with $6 in it initially.

00:07 Player 1, alice moves first, then player 2, bob, then player 1 again, and finally player 2 again.

00:14 At each player's turn to move, they have two possible actions, grab g or share s.

00:20 If the player grabs, they get 2 thirds of the current pot of money, and the other player gets one third of the pot and the game ends.

00:27 If they share, then the size of the pot is multiplied by three halves, and the next player gets to move.

00:35 At the last stage in which bob moves, if he chooses to share, then the pot is still multiplied by three halves, and bob gets one -third of the pot, and alice gets two -thirds of the pot.

00:49 A, model this as an extensive form game and draw the game tree.

00:53 Is it a game of perfect or imperfect information? b, how many terminal nodes does the game tree have? how many information sets are there? what about the number of subgames? c, please use backward induction to find the subgame perfect nash equilibrium.

01:13 It is given that there are two players and there are two possible actions to choose from, grab or share and we're starting with $6 in the pot.

01:23 So let's start with this game tree.

01:31 So the game tree is going to be as follows.

01:35 We start with alice because she moves first.

01:37 She can share or she can grab if she grabs it's split with four and two one person gets four dollars the other person gets two dollars and now it's bob's turn so bob's going to go next and i'm actually going to pick a different color so we can distinguish where bob is in this tree so bob is going to be blue and bob's choices are to share or grab and if he grabs at this point it would be split with $3 .6.

02:32 And now it is alice's turn again and alice's choices are once again to share or grab.

02:47 And if she grabs at this juncture, we have 9 and 4 .5 for the split.

02:53 And now it's bob's turn again and his choices are to share or grab.

03:04 And that split is going to be 6 .75 and 13 .5.

03:13 This is a game of perfect information...

Question

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