A conjecture about Nash Equilibria in multiplayer games involving card drafting A deck of $N$ cards is used to play a 4-player game. The game begins with each player being randomly dealt 7 cards from the deck. They then take turns according to a set of rules, after which a single winner is crowned. The exact rules of the game are complex and not worth explaining in detail.

Certain cards in the deck are better than others, so the players decide to institute a pre-game draft to even out the variance. In the draft, each player is randomly dealt 7 cards face-down from the deck. Each player looks at his cards, chooses one to keep, and passes the remaining 6 cards face-down to the player to his left. After each player has done this, the process is repeated, this time each player choosing from the 6 cards received from his right, choosing one to keep, and then passing 5 to the player to his left. This is repeated until each player has chosen exactly 7 cards. They then proceed with the actual game.

Now suppose that the draft-version of this game has a unique Nash Equilibrium strategy, and let us imagine that 4 copies of this strategy play against each other. We can define two functions of cards that might be representative of their strength:

Expected-draft-position: The expected-draft-position of a card $c$, $\mathrm{EDP}(c)$, is the expected position in which $c$ is drafted, given that $c$ was dealt out in the draft. For instance, if a card $c$ is always the first card drafted, then $\mathrm{EDP}(c)=1$. If it is always the last card drafted, then $\mathrm{EDP}(c)=7$. Note that the average $\mathrm{EDP}$ over all $N$ cards must be 4.

Win-probability: The win-probability of a card $c$, $\mathrm{WP}(c)$, is the probability that the player that drafted $c$ will win the game, given that $c$ was dealt out in the draft. Note that the average $\mathrm{WP}$ over all $N$ cards must be 0.25.

Conjecture: $\mathrm{EDP}(x) &lt; \mathrm{EDP}(y)$ iff $\mathrm{WP}(x) &gt; \mathrm{WP}(y)$ for all cards $x, y$.

Question: Is this conjecture true? If not, are there some "minimal" assumptions we can make about the game rules so that it becomes true?

A little background: I am part of an online board game community, and the above two statistics are computed empirically from our game logs, which involve imperfect human players rather than Nash Equilibrium strategies. For certain pairs of cards, my conjecture does not hold, and I am personally convinced that this must indicate that players are drafting sub-optimally. I would like to be certain of this conviction via mathematical proof.
After discussion with bjorn, we arrived at the following counterexample. However, the part of my question where I ask about a minimal set of conditions to make the conjecture true remains open. 



Counterexample

Consider a deck with N=28 cards, so that every card is dealt out in every game. The cards, $c_1, c_2, \ldots, c_{28}$, are positive real values, with $c_k = 1.0+k\epsilon$ for some positive $\epsilon \ll 1$.

Say player $i$ has drafted cards $S_i$. The game works as follows: the winner is randomly chosen, with player $i$’s chances of winning proportional to $p(S_i)$, where 

$$
P(S) = 
\begin{cases}
    10^{100},&amp; \text{if } c_1, c_{28} \in S\\
    \prod_{c\in S} c,              &amp; \text{otherwise}
\end{cases}
$$

The Nash Equilibrium strategy is clear: draft $c_1$ if necessary. Else, draft the highest available card. Clearly, at equilibrium, the $10^{100}$ bonus will never be obtained.

It is easy to see that $\mathrm{EDP}(c_1) &lt; \mathrm{EDP} (c_2)$ and that $\mathrm{WP}(c_1)  &lt; \mathrm{WP}(c_2)$.
I don't think the answer can be fixed because if a player knows that the player on his right used a certain strategy and that all the players use the same thought process as him, he will improve his strategy accordingly. Over time, the previous player will get to know of his strategy and modify his accordingly, so will all the players after him. This will keep on repeating till the players realize that they can't outwit the other players. They will use the best strategy that works independently of the other players' decisions (and probably keep the best card with them every time).

I don't know about the math, but from a logical point of view, this should be the answer. It is the same reason why there can never be a perfect strategy that works against itself in card games like no-limit holdem poker.