[superprismatic]

Two players, A and B, play the following

game: Each secretly chooses a real

number and passes his choice to a judge.

Neither A nor B gets to see the other's

choice of number. The judge then

multiplies the two numbers and reveals

the most significant (non-zero) digit of

the result. If this digit is 1, 2, or

3, player A wins $1 and B wins nothing.

If this digit is 4, 5, 6, 7, 8, or 9,

player B wins $1 and A wins nothing.

If they play N games, each using an

optimal strategy, what is the expected

winnings for each player? Why?

[superprismatic]

I don't have time to go into it right now. Suffice it to say that taking logs (base 10) will straighten out those curves and make the strategies clear and gives A a probability of winning of just over 60%.

Sorry for the late response. I didn't internet access for the last few days. Here is how I approached the problem,

I first summarize the game with a win/loss map that plots player A/player B numbers and the corresponding outcome. As noted before, it is only necessary to consider numbers between [1, 10). In the following map, the x-axis represent player B's strategy, and y-axis player A's. The red areas are regions where player B would win. The areas are separated by three lines whose equations are displayed on the map,

So the strategy is as follows. Suppose we can only pick 5 numbers as player B, how much red 'areas' can we cover using only 5 points? I started out by picking x1 = 10 as the first number, as it can win over all numbers inside between 4 and 10 from player A (see graph below for illustration). I then project horizontal line (y=4) and see where it meets with the next red region (xy = 10), and then choose that as my next point (x2 = 2.5). The number x2 can beat numbers between 1.6 and 4 from player A. I then draw a horizontal line (y=1.6) and compute to see where it crosses the line (xy = 10). That gives me my third point x3 = 6.25. The graph below carries out the procedure for x1 to x5.

Notice that after these 5 numbers, almost every single number on the y-axis can be between by 2 out of 5 of player B's number. I'm saying 'almost' because the fifth number x5 = 3.90625 has a little bit of white areas under neath it, so the numbers interval (1, 1.024) on the y-axis is only covered once by these 5 numbers. This is not much of a problem, since we can simply add in 5 more numbers using the same procedure with a result that almost every number on the y-axis can be beaten by 4 out of 10 of player B's numbers. I cut the procedure short after 13 numbers (with the resulting winning rate 5 out of 13), but we can repeat this procedure for a longer time and get a winning rate that is approaching 2/5.

This is a wonderfully crafted problem, thanks for posting it. I would love to hear about your strategy on this problem.

[bushindo]

I don't have time to go into it right now. Suffice it to say that taking logs (base 10) will straighten out those curves and make the strategies clear and gives A a probability of winning of just over 60%.

That's a marvelous approach! Thanks for sharing this great puzzle and the elegant solution.

Edited January 19, 2011 by bushindo

[superprismatic]

A's best strategy is to pick a random x which

is uniformly distributed in the interval [0,1),

and choose 10^x as his secret real number.

This strategy will insure that he wins about

60% of the time (log₁₀(4), to be precise).

B's best strategy is to choose his number the

same way, but he will win only about 40% of

the time (1-log₁₀(4)).

This can be seen by noticing, as Quantum.Mechanic

has, that the number we pick can, without loss

of generality, be chosen in the interval [1,10).

Then we notice that, if a and b are A's and B's

choices respectively, A wins when 1 ≤ ab < 4 or

10 ≤ ab < 40. Then we take log₁₀ of everything,

thereby converting this to an additive instead

of a multiplicative problem. Analysis is

pretty straightforward from this point on.