A and B play the following game: A writes downeither

Chapter 4, Problem 4.24

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A and B play the following game: A writes downeither number 1 or number 2, and B must guesswhich one. If the number that A has written downis i and B has guessed correctly, B receives i unitsfrom A. If B makes a wrong guess, B pays 34unit toA. If B randomizes his decision by guessing 1 withprobability p and 2 with probability 1 p, determinehis expected gain if (a) A has written downnumber 1 and (b) A has written down number 2.What value of p maximizes the minimum possiblevalue of Bs expected gain, and what isthis maximin value? (Note that Bs expected gain depends not only on p, but also on whatA does.)Consider now player A. Suppose that she alsorandomizes her decision, writing down number 1with probability q. What is As expected loss if (c)B chooses number 1 and (d) B chooses number 2?What value of q minimizes As maximumexpected loss? Show that the minimum of As maximumexpected loss is equal to the maximum of Bsminimum expected gain. This result, known as theminimax theorem, was first established in generalityby the mathematician John von Neumann andis the fundamental result in the mathematical disciplineknown as the theory of games. The commonvalue is called the value of the game to player B.