Question

In a quiz show a uniformly random integer r between 1 and 10 is generated. Another, independent such random numbers will then be generated, but before that happens, you are invited to guess whether s will be greater than or less than r. If you are correct, then you win s pounds. If you lose (or if s = r) then you win nothing. (i) Clearly if r = 1 you should guess that s will be larger. And if r = 10 you should guess that s will be smaller. At which value of r should your strategy change from guessing s will be larger to guessing it will be smaller? (Your aim, as always, is to maximise your expected gain.) (ii) Suppose now that the range of possible values for r and s is 1, ... , N. Then in the limit as N tends to infinity the change of strategy should happen at a value of r of approximately N/k. Find the value of k.

Answer 1

(i)

Our aim is to maximize the expected gain.

If r=1, expected gain for guessing s is large = P(s=1)*0+P(s=2)*2+...+P(s=10)*10

If r=1, expected gain for guessing s is small = P(s=1)*0+P(s=2)*0+...+P(s=10)*0

Since s is a uniform random number between 1 and 10, P(s=i)=1/10 for all i in [1,10]

Therefore, expected gain for guessing s is large= 1/10*(2+3+...+10)

  =5.4 pounds

Also, expected gain for guessing s is small= 0

Similarly, if r=2, expected gain for guessing s is large = P(s=1)*0+P(s=2)*0+P(s=3)*3+...+P(s=10)*10

=1/10*(3+4+...+10)

=5.2 pounds

Also, expected gain for guessing s is small= P(s=1)*1+P(s=2)*0+P(s=3)*0+...+P(s=10)*0

=0.1 pound

Proceeding similarly, expected gain at each value of r can be summarized as below:

r	expected gain for guessing s is large	expected gain for guessing s is small
1	5.4 pounds	0
2	5.2 pounds	0.1 pounds
3	4.9 pounds	0.3 pounds
4	4.5 pounds	0.6 pounds
5	4 pounds	1 pounds
6	3.4 pounds	1.5 pounds
7	2.7 pounds	2.1 pounds
8	1.9 pounds	2.8 pounds
9	1 pounds	3.6 pounds
10	0 pounds	4.5 pounds

So if r is less than or equal to 7, then guess s is large and if r is greater than 7, guess s is small.

So, the value of r at which our strategy should change from guessing s is large or small is 7.

(ii)

Suppose the range of possible values of r and s is 1,2,...N.

Proceeding as in above case, we can see that if r=i where i is in [1,N], the expected gain in guessing s is large can be formulated as 1/N*[(i+1)+(i+2)+...+N]

and the expected gain in guessing s is small can be formulated as 1/N*[1+2+...+(i-1)]

So, our strategy changes for  1/N*[1+2+...+(i-1)] > 1/N*[(i+1)+(i+2)+...+N]

i.e, 1+2+...+(i-1) > (i+1)+(i+2)+...+N

L.H.S of above equation is sum of first i-1 natural numbers = (i-1)*i/2

R.H.S of above equation is (sum of first N natural numbers) - (sum of first i natural numbers) = N*(N+1)/2 - i(i+1)/2

So, i is such that (i-1)*1/2 > N*(N+1)/2 - i(i+1)/2

On solving, i= [N*(N+1)/2]^1/2

In the limit as N tends to infinity the change of strategy should happen at r = N/k

where N/k = [N*(N+1)/2]^1/2

Solving for k, k=[2*N/(N+1)]^1/2