You enter a special kind of chess tournament, whereby you play one game with eac

Question

You enter a special kind of chess tournament, whereby you play one game with each of three opponents, but you get to choose the order in which you play your opponents. You win the tournament if you win two games in a row. You know your probability of a win against each of the three opponents. What is your probability of winning the tournament, assuming that you choose the optimal order of playing the opponents? (Assume that in any game someone wins and someone loses; i e., there is no draw.) Note that the underlying probability model is not explicitly given so you will have to define it. Give reasons why your model is a reasonable one.

Explanation / Answer

Firstly we assume that each game is independent of each other. Let the players be A, B and C. Let WA , WB, and WC be the event of a win against A, B and C respectively. Similarly define LA , LB and LC as the event of a loss against A, B and C respectively. Let the probabilities of winning against A, B and C be pA , pB and pC respectively.

Now you can win the tournament if you win two consecutive games, so either it is WAWBWC or WAWBLC or LAWBWC . Now since each game is independent, then P(WAWBWC) = P(WA)P(WB)P(WC) = pApBpC, P(WAWBLC) = P(WA)P(WB)P(LC) = pApB(1 - pC) , P(LAWBWC) = P(LA)P(WB)P(WC) = (1 - pA)pBpC.

So the probability of winning is = pApBpC + pApB(1 - pC) +(1 - pA)pBpC = pB(pA + (1 - pA ) pC)

This order (play A first, then B then C) is optimal if the probabilities of other orders are less than the given order, ie.

pB(pA + (1 - pA ) pC) >= pA(pB + (1 - pB ) pC) .......(1) (B is played first, A is played second and C third)

pB(pA + (1 - pA ) pC) >= pC(pB + (1 - pB ) pA) .......(2) (B is played first, C is played second and the A is third)

(Note that it only matters who plays second, because the probability in other cases, ie. when a player is first or if he is third, are the same)

From (1) we get pB>=pA and from (2) we get pB>=pC . Thus we should always play an opponent, against whom you have the maximum probability of winning, second, thereby maximising our chance of winning the tournament.

Hence our model is reasonable model as the only assumption it makes is of indipendence which in most cases people will agree is a reasonable assumption. Also we have shown that the strategy of playing second the weakest player will maximise the chances of winning the tournament.

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