GAME THEORY **write out the complete strategy sets(follow \"derive the strategy

Question

GAME THEORY
**write out the complete strategy sets(follow "derive the strategy set for each player" and ignore the part that says "or, alternatively state a representative strategy for a player")**

Exercises 8. Consider the extensive form game below. The top number at a terminal node is player 1's payoff, the second number is player 2's payoff, the third num- ber is player 3's payoff, and the bottom number is player 4's payoff. a. Derive the strategy set for each player or, alternatively, state a represen- tative strategy for a player b. Derive all subgame perfect Nash equilibria. A 2 3 2 4 al b, A 0 1 3 3 0 3 24 3 2 1 3 4 52

Explanation / Answer

a).

Consider the given problem, here there are 4 players all of them have 2 possible strategies. So, player1(P1) have strategy set {a1, b1}, player2(P2) have strategy set {a2, b2}, player3(P3) have strategy set {a3, b3}, player4(P4) have strategy set {a4, b4}.=> “Pi” have strategy set {ai, bi}.

b).

Now, to derive all the possible SPNE we need to follow back ward induction method.

P1 will play “b1”, “P2” will play “a2” and “P3” will play “a3”, here P4 will choose “a4”, since 3 > 2. Now, if P1 will play “b1”, “P2” will play “a2” and “P3” will play “b3”, here P4 will choose “b4”, since 3 > 1.

So, under this situation P3 will choose “a3”, since in the former case “P3’s” pay-off was 1, in the latter one it’s pay-off is 0, so “P3” will choose “a3”, if P1 will play “b1”, “P2” will play “a2”, and P4 will choose “a4” and will get (3, 2, 1, 3) as pay-off.

P1 will play “b1”, “P2” will play “b2” and “P3” will play “a3”, here P4 will choose “b4”, since 5 > 4. Now, if P1 will play “b1”, “P2” will play “a2” and “P3” will play “b3”, here P4 will choose “a4”, since 2 > 1.

So, under this situation P3 will choose “b3”, since in the former case “P3’s” pay-off was 2, in the latter one it’s pay-off is 4, so “P3” will choose “b3”, if P1 will play “b1”, “P2” will play “b2” and P4 will choose “a4” and will get (0, 3, 4, 2) as pay-off.

Now, if we compare both these situation then we will see that P2 will get more in the latter case, So, if P1 will choose “b1”, then “P2” will choose “b2”, then P3 will choose “b3” and P4 will choose “a4”, the game will end up with pay-off (0, 3, 4, 2).

Now, assume that if “P1” will choose “a1”, then P2 have 2 option either “a2” or “b2”, so here “P2” will choose “b2” since 4 > 2. So, if “P1” will choose “a1”, then “P2” will choose “b2” and the game will end up with pay-off (2, 4, 2, 3).

Now, “P1” will decide between “a1” or “b1”, if “P1” will choose “a1” then the final outcome will be (2, 4, 2, 3), on the other hand if “P1” will choose “b1” then the final outcome will be (0, 3, 4, 2). So, “P1” will choose “a1” since 2 > 0.

So, the SPNE of this game is (P1=a1, P2=b2)=(2, 4, 2, 3).

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