I am analysing a game based on an extension of the Monty Hall problem and have come across an optimization problem that is just not clicking. I do not currently understand fully whether I should be thinking about it as a linear program or a quadratic program, or something else entirely.
The Game
We have a player, a host and $N$ indistinguishable, randomly arranged doors, where $N \ge 3$.
The $i^{th}$ door conceals a prize of value $\omega_i$, with $\omega_{i+1} \ge \omega_i$.
Before playing, the host nominates a door (not door $N$) to be door $X$.
The player selects a door.
- If the player selects door $N$, the host opens a door* with probability $m_N$.
- If the player selects door $X$, the host opens a door with probability $m_X$.
- If the player selects some other door $k$, the host opens a door with probability $b$.
*(N.B. The door opened by the host cannot be $N$, $X$, or the door chosen by the player. This also means that for $N=3$, $b$ must be $0$.)
Once the host has opened a door (or declared that a door will not be opened), the player is allowed to "twist" to another unopened door instead of their initial selection.
The player is able to develop a strategy wherein they twist with probability $c$ when the host opens a door, and with probability $n$ when a door is not opened. In developing the strategy, $c$ and $n$ will only take the values $0$ or $1$.
The player's final choice, door $F$, is opened and the player wins the prize $\omega_F$.
The Problem
I have generated an expression for the expected prize winnings $\overline{\omega}$ in terms of $m_N, \ m_X, \ b, \ \omega_N, \ \omega_X, \ \sum_{k \ne X}^{N-1} \omega_k$ and the player's strategy $c, \ n$.
$$\begin{multline}{
\overline{\omega} = \frac{1}{N} \left( m_N \left[ (n-c)\omega_N + \left( \frac{c}{N-2} - \frac{n}{N-1} \right) \omega_X + \left( \frac{(N-3)c}{(N-2)^2} - \frac{n}{N-1} \right) \sum_{k \ne X}^{N-1} \omega_k \right] \right. \\
\left. + m_X \left[ \left( \frac{c}{N-2} - \frac{n}{N-1} \right) \omega_N + (n-c) \omega_X + \left( \frac{(N-3)c}{(N-2)^2} - \frac{n}{N-1} \right) \sum_{k \ne X}^{N-1} \omega_k \right] \right. \\
\left. + b \left[ \left( c - \frac{(N-2)n}{N-1} \right)(\omega_N + \omega_X) + 2\left(\frac{n}{N-1} - \frac{c}{N-2}\right) \sum_{k \ne X}^{N-1} \omega_k \right] + \left[\omega_N + \omega_X + \sum_{k \ne X}^{N-1} \omega_k \right] \right) }\end{multline}$$
For each strategy $(c,n)$, I want to minimize $\overline{\omega}$ while keeping the probability that the host opens a door $\mathbb{P}(open) = \frac{m_N + m_X + (N-2)b}{N}$ above some value $\beta$. Ultimately, I want an expression (or expressions) that I can use to work out the minimum $\overline{\omega}$, given the prize values $\omega_i$, the player's strategy $(c,n)$ and $\beta$.
Minimize $$\overline{\omega}\left(m_N,m_X,b,\omega_N,\omega_X,\sum_{k \ne X}^{N-1}\omega_k,c,n\right)$$
Subject to:
$$\begin{align}{
m_N, m_X, b \le 1 \\
\omega_X \le \omega_N\\
\sum_{k \ne X}^{N-1} \omega_k \le (N-2)\omega_N \\
\frac{m_N + m_X + (N-2)b}{N} \ge \beta
}\end{align}$$
Attempts So Far
Until now, I have been thinking of $m_N, \ m_X, \ b$ as the decision variables in a linear programming problem. (One that I have been attacking using simplex algorithm.) This means I have to consider many cases: each player strategy, and whether $w_X$ is less or greater than various multiples of $\sum_{k \ne X}^{N-1}\omega_k$.
However, if I were to treat this as a quadratic program and include the $\omega_i$ as decision variables, I believe I would find that $\overline{\omega}$ is trivially minimized by setting all the prizes to $0$ unless I find some other constraint for the prizes.
I am looking for suggestions as to what I might do next to make progress, as I would rather not have to work out every possible case of the linear program and attend to them individually. Though, of course I will if I must.