Watch the following Golden Balls video at https://youtu.be/S0qjK3TWZE8 before reading the rest of the question.

Let's call the contestant on the left player 1, and the contestant on the right player 2. Player 2 follows the rather unconventional strategy of saying "no matter what, I will choose steal, but if you choose split, I will split the money with you afterwards". The host reminds the contestants that player 2 is under no legal obligation to split the money after the show. In the end, the contestants both choose split.

Let's analyze why this seemingly crazy strategy makes sense. We will model this by assuming that player 2 has three types:

? Steal but share money afterwards (type T) with probability ?. If type T, player 2 will always play steal, but will split the money afterwards if player 1 plays split.

? Always split (type P) with probability ?. If type P, player 2 will always play split.

? Normal (type N) with probability 1 ? ? ? ?. If type N, player 2 maximizes his winnings.

We can represent the game as in Figure 1 (attached). In this representation, we assume that player 1 cares only about maximizing his winnings. To make things simpler, we do not model player 2 types T and P directly - I have just put the payoffs corresponding to their strategy into the game tree. Hence, we only model the choice of player 2 if he is type N. Notice that player 1 does not observe player 2's type. The first payoff is that of player 1, the second that of player 2. I have normalized the payoff from winning to 100. Suppose ? > 0 and ? > 0.

Acutal questions:

(a) Does this game have complete or incomplete information? Does it have perfect or imperfect information? Write up the strategy sets of the two players.

(b) Suppose player 2 plays Steal when he is type N such that s2 = Steal. Let player 1's expected payoff given the strategies and beliefs be denoted u1(s1, s2; ?, ?). Show that u1(Split, Steal; ?, ?) = 50? + 50?, and find u1(Steal, Steal; ?, ?).

(c) Find the parameter values ? and ? such there is a perfect Bayesian Nash equilibrium (PBE) where player 1 plays Split and player 2 type N plays Steal. In the video, he claims that he wants to split, but here we make it simple by just assuming that he maximizes winnings.

(d) Why can player 2 not just say "I'm type P, I will always play split." In other words, why is it important in the video that player 2 introduces the possibility of type T, even if he turns out to be type P.

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