Problems

In a game for two players, a fair coin is tossed repeatedly. Each player is assigned a sequence of heads and tails and the player whose sequence appears first wins. Four players, \(A\), \(B\), \(C\) and \(D\) take turns to play the game. Each time they play, \(A\) is assigned the sequence TTH (i.e.~Tail then Tail then Head), \(B\) is assigned THH, \(C\) is assigned HHT and \(D\) is assigned~HTT.

1. \(A\) and \(B\) play the game. Let \(p_{\mathstrut\mbox{\tiny HH}}\), \(p_{\mathstrut\mbox{\tiny HT}}\), \(p_{\mathstrut\mbox{\tiny TH}}\) and \(p_{\mathstrut\mbox{\tiny TT}}\) be the probabilities of \(A\) winning the game given that the first two tosses of the coin show HH, HT, TH and TT, respectively. Explain why \(p_{\mathstrut\mbox{\tiny TT}} = 1\,\), and why $p_{\mathstrut\mbox{\tiny HT}} = {1 \over 2} \, p_{\mathstrut\mbox{\tiny TH}} + {1\over 2} \, p_{\mathstrut\mbox{\tiny TT}}\,$. Show that $p_{\mathstrut\mbox{\tiny HH}} = p_{\mathstrut\mbox{\tiny HT}} = {2 \over 3}$ and that \(p_{\mathstrut\mbox{\tiny TH}} = {1\over 3}\,\). Deduce that the probability that A wins the game is \({2\over 3}\,\).
2. \(B\) and \(C\) play the game. Find the probability that \(B\) wins.
3. Show that if \(C\) plays \(D\), then \(C\) is more likely to win than \(D\), but that if \(D\) plays \(A\), then \(D\) is more likely to win than \(A\).

In the game of ``Colonel Blotto'' there are two players, Adam and Betty. First Adam chooses three non-negative integers \(a_{1},a_{2}\) and \(a_{3},\) such that \(a_{1}+a_{2}+a_{3}=9,\) and then Betty chooses non-negative integers \(b_{1},b_{2}\) and \(b_{3}\), such that \(b_{1}+b_{2}+b_{3}=9.\) If \(a_{1} > b_{1}\) then Adam scores one point; if \(a_{1} < b_{1}\) then Betty scores one point; and if \(a_{1}=b_{1}\) no points are scored. Similarly for \(a_{2},b_{2}\) and \(a_{3},b_{3}.\) The winner is the player who scores the greater number of points: if the socres are equal then the game is drawn. Show that, if Betty knows the numbers \(a_{1},a_{2}\) and \(a_{3},\) she can always choose her numbers so that she wins. Show that Adam can choose \(a_{1},a_{2}\) and \(a_{3}\) in such a way that he will never win no matter what Betty does. Now suppose that Adam is allowed to write down two triples of numbers and that Adam wins unless Betty can find one triple that beats both of Adam's choices (knowing what they are). Confirm that Adam wins by writing down \((5,3,1)\) and \((3,1,5).\)

A and B play a guessing game. Each simultaneously names one of the numbers \(1,2,3.\) If the numbers differ by 2, whoever guessed the smaller pays the opponent £\(2\). If the numbers differ by 1, whoever guessed the larger pays the opponent £\(1.\) Otherwise no money changes hands. Many rounds of the game are played.

1. If A says he will always guess the same number \(N\), explain (for each value of \(N\)) how B can maximise his winnings.
2. In an attempt to improve his play, A announces that he will guess each number at random with probability \(\frac{1}{3},\) guesses on different rounds being independent. To counter this, B secretly decides to guess \(j\) with probability \(b_{j}\) (\(j=1,2,3,\, b_{1}+b_{2}+b_{3}=1\)), guesses on different rounds being independent. Derive an expression for B's expected winnings on any round. How should the probabilities \(b_{j}\) be chosen so as to maximize this expression?
3. A now announces that he will guess \(j\) with probability \(a_{j}\) (\(j=1,2,3,\, a_{1}+a_{2}+a_{3}=1\)). If B guesses \(j\) with probability \(b_{j}\) (\(j=1,2,3,\, b_{1}+b_{2}+b_{3}=1\)), obtain an expression for his expected winnings in the form \[ Xa_{1}+Ya_{2}+Za_{3}. \] Show that he can choose \(b_{1},b_{2}\) and \(b_{3}\) such that \(X,Y\) and \(Z\) are all non-negative. Deduce that, whatever values for \(a_{j}\) are chosen by A, B can ensure that in the long run he loses no money.