Best way to create sample set of numbers 1 to 5 to minimize expected value when betting on outcome of sample

Question

You and I play the following game: You create a deck of as many cards as you want, and on every card you write one of the integers from $1$ to $5$, e.g. your deck could be just $\{1,2,3,4,5\}$ or $\{1,1,1,1,2,3,3,3\}$ or whatever you want, as long as it only consists of the integers from $1$ to $5$. Then, I have a look at the deck, will select one number to bet on, and one random card will be chosen from your deck. If that number matches my bet, you pay me that that number. How do you create a deck in order to minimize your losses, given that I play optimally?\

The best I figured out was $\{1,1,1,1,1,2,2,3,4,5\}$, giving you an expected loss of $\frac{1}{2}$ if I bet on $1$ or $5$ and less for the other numbers, but I'm quite sure that this is not optimal but also don't know how to proceed from there. Maybe there is a solution that gives me the same expected value no matter on which number I bet and is thus better for you?

Answer 1

Suppose that a fraction $p_i$ of the cards are labelled $i$, for $1\leq i\leq 5$.

The crucial insight is that your best strategy will have $$1p_1=2p_2=\dots=5p_5,$$ so that the opponent is indifferent to which card they guess. You can then see that the expected value is $\frac{60}{137}$.

Indeed, if we don't follow this distribution, then some $j$ would have $jp_j>\frac{60}{137}$, so the opponent could always guess card $j$ and have a higher EV.