Question

Please do D,E,G and H

4. Peter and Paul will play a game. Peter has two possible choices: OPASS or ORUSH; while Paul has two possible choices: DPASS or DRUSIH. For each combination of choices of Peter and Paul, there is a resulting GAIN for Peter. These GAIN values are given in the following two-by-two table. For example, if Peter chooses OPASS while Table 1: Table of gain values for Peter for combinations of choices by Peter and Paul. Value of Peter's GAIN Peter Chooses OPASS ORUSH Paul chooses DPASS 2 Paul chooses DRUSH3 Paul chooses DRUSH, then Peter will gain the value of 3. Peter's goal is to maximize his gain; while Paul's goal is to minimize Peter's gain. Peter and Paul will choose their actions independently of each other. Peter will choose OPASS with probability o and ORUSII with probability 1o; whereas, Paul will choose UPASS with probability β and DRUSII with probability 1 -β. INote: Both α and ß are between 0 and 1, inclusive.! inclusive .เ α and β are between o and 1, n? ain? (c) show that in general, Peter's expected gain, as a function of (α, β) and denoted by V(α, β), is given by where we denoted by G the random variable representing Peter's gain. (d) Plot this function in 3-dimensional space. Describe the shape of the surface. Is it shaped like a horse saddle? (e) The saddlepoint is the solution of the two equations -V(α, β)-0 and V(a, β)-0. Demonstrate that the saddlepoint of V(α, β) is given by 4 2 Then locate this point in the plot you created in item (d). (g) At this saddlepoint (α*,P), what is Peter's expected gain? (h) Verify that the following equalities hold true: y(a",β.) maxi-mini V (α,β) minimax V(α,β) ının max . Rernark: The sudd1 point (o". β. , which represents the optimal probabilities for Peter and Paul, is called the Nash Equilibrunn Point and V(a",β*) is called the Value of the Game. John Nash got the Nobel Prize in Economics for discovering the more general result, but the essence of what he got was what you did in this problem! Johın Nash was the subject of the movie Beautiful Mind

Answer 1

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