# Infinite Cards to choose from, how to minimize their expected value?

There are infinite cards numbered 1-5. You construct a deck. Someone guesses a value and draws a card. If they guessed correctly, they get the value. What composition of the deck minimizes their expected value?

The answer is purportedly: Keep expected value constant so the value and number of each card are inversely proportional

But I don't get what they mean. Maybe I have misunderstood the question.

Surely making a deck of all 1s, would minimise the EV?

Is it asking to minimize $E=\sum ^{5}_{i=1}ip_{i}$

such that

$\sum ^{5}_{i=1}p_{i}=1$

• If all cards are $1$ then the player guesses $1$ and guarantees $1$. If $\frac 13$ are $2$ and $\frac 23$ are $1$ then either guess has expectation $\frac 23$, just to toss one alternative out....so your strategy is sub-optimal. – lulu Jun 12 '18 at 17:45
• No, you want to minimize $M\equiv \max\{ip_i\mid 1\le i\le 5\}$ subject to $\sum_{i=1}^5 p_i=1$. The minimum occurs when $ip_i=jp_j$ for all $1\le i\le j\le 5$. If we had some $ip_i>jp_j$, then you could decrease $p_i$ and increase $p_j$ without increasing $M$. – Mike Earnest Jun 12 '18 at 17:56
• @lulu, right well I wasn't sure if the person knew what deck you chose, they just know it contains cards 1-5 – Tinatim Jun 12 '18 at 19:11
• Well, I agree the problem is somewhat vaguely worded. Still, the interesting problem comes when you assume that both parties have full information and both make optimal moves. – lulu Jun 12 '18 at 19:22
• @MikeEarnest Thanks, I assume that is the correct answer, assuming both parties have full information. – Tinatim Jun 12 '18 at 19:23