Counting on numbers

1. Here is a very simple game. You come up with a number and I come up with a number. If I come up with the higher number, I win; otherwise you win. You go first. Call this ‘The Very Simple Game’. Few would play it if they had to go first and many if they are guaranteed to go second.

2. Here is another one. You come up with a number n and I come up with a number m. If m times 1/n > 1, then I win; if not, then you win. You go first. Call this ‘Still The Very Simple Game’. Since I win just in case m is greater than n, this game collapses into The Very Simple Game. Few people will play it if they have to go first.

3. Here is a third one. Let us call it ‘Looks Like A Tricky Game’. Suppose you have a choice between exactly two actions:

• (A) keep holding on to some thing t (with no further relevant changes);

• (B) hand t over to me and accept my offer to give you, with some probability p (1/n), some thing with (positive) utility m (with no further relevant changes).

Suppose, you are a maximizer of expected utility with …