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One point away — page 2

Constructions that work perfectly except at a single exceptional place, and what is done about it.
A series that gets better, then worse. The error of Euler's series against the number of terms kept, at x = 0.05 and 0.1 and 0.2. At 0.05 the error falls to 1.1e-8 at 20 terms and then climbs without limit. At 0.1 the error falls to 1.8e-4 at 10 terms and then climbs without limit. At 0.2 the error falls to 1.8e-2 at 5 terms and then climbs without limit. Analysis

A series that converges nowhere

Expand Euler's integral in powers of x and the coefficients are the factorials, so the series converges at no x but nought. Stopped at its smallest term it still computes the integral to within about e^(−1/x) — and every term added after that makes the answer worse.

 Announcing a mixture is worth 5/3 more than any equilibrium. The leader's payoff against the probability it announces for its first action, for a leader with a dominant action that is better off not being seen to play it, with the follower's reply switching where the follower is indifferent. The best announcement is worth 11/3; the best equilibrium of the simultaneous game is worth 2. Applied

Worth more for being seen first

Moving first sounds like a disadvantage, since the other side gets to see the move and answer it. When the move is a mixture that is announced and believed, it is never a disadvantage, it is worth exactly nothing in a game of pure conflict, and in other games it is worth more than any equilibrium — sometimes by announcing an action that would never be played in secret.

 Up to 4 choices among 60, and the thresholds that nest. For 60 candidates in random order and 1, 2, 3, 4 acceptances, the chance of ending with the best among those accepted — 37.3%, 60.0%, 74.3%, 83.5% — and where along the sequence the rule starts accepting for each number of choices in hand. Probability

The thresholds that nest

Allow a second acceptance in the secretary problem and the chance of holding the best rises from about 37 per cent to about 59. The best rule is still a threshold — but one threshold for each number of choices still in hand, the earlier ones starting sooner, and each additional choice buying less than the one before.

 Add the odds from the end: start at event 5 of 12. The odds of 12 independent events with chances 1/k — the secretary problem, the sum of the odds from the end reaching one at event 5, and the chance of stopping on the last success from each starting point, highest at 39.6%. Probability

Add the odds from the end

Watch a sequence of independent events and try to stop exactly on the last one that happens. Add up the odds of the events from the end backwards until the total reaches one, and stop at the first success from there. That rule is the best possible for any probabilities whatever, and the secretary problem is the special case in which the chances are one over the position.

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